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MeasureTheory.IsStoppingTime.measurableSet_min_iff ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π s : Set Ω ⊢ MeasurableSet s ↔ MeasurableSet s ∧ MeasurableSet s ** rfl ** Qed
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MeasureTheory.IsStoppingTime.measurableSpace_min_const ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ i : ι ⊢ IsStoppingTime.measurableSpace (_ : IsStoppingTime f fun ω => min (τ ω) i) = IsStoppingTime.measurableSpace hτ ⊓ ↑f i ** rw [hτ.measurableSpace_min (isStoppingTime_const _ i), measurableSpace_const] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_inter_le_const_iff ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ i}) ↔ MeasurableSet (s ∩ {ω | τ ω ≤ i}) ** rw [IsStoppingTime.measurableSet_min_iff hτ (isStoppingTime_const _ i),
IsStoppingTime.measurableSpace_const, IsStoppingTime.measurableSet] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι ⊢ (∀ (i_1 : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ i_1})) ↔ (∀ (i_1 : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ i_1})) ∧ MeasurableSet (s ∩ {ω | τ ω ≤ i}) ** refine' ⟨fun h => ⟨h, _⟩, fun h j => h.1 j⟩ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι h : ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ i_1}) ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ i}) ** specialize h i ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι h : MeasurableSet (s ∩ {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ i}) ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ i}) ** rwa [Set.inter_assoc, Set.inter_self] at h ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_le_stopping_time ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π ⊢ MeasurableSet {ω | τ ω ≤ π ω} ** rw [hτ.measurableSet] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π ⊢ ∀ (i : ι), MeasurableSet ({ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ i}) ** intro j ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} ⊢ MeasurableSet ({ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j}) ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} ⊢ MeasurableSet ({ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j}) ** refine' MeasurableSet.inter _ (hτ.measurableSet_le j) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} ⊢ MeasurableSet {ω | min (τ ω) j ≤ min (π ω) j} ** apply @measurableSet_le _ _ _ _ _ (Filtration.seq f j) _ _ _ _ _ ?_ ?_ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ⊢ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω ⊢ ω ∈ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} ↔ ω ∈ {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} ** simp only [Set.mem_inter_iff, Set.mem_setOf_eq, min_le_iff, le_min_iff, le_refl, and_true_iff,
and_congr_left_iff] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω ⊢ τ ω ≤ j → (τ ω ≤ π ω ↔ (τ ω ≤ π ω ∨ j ≤ π ω) ∧ (τ ω ≤ j ∨ True)) ** intro h ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω h : τ ω ≤ j ⊢ τ ω ≤ π ω ↔ (τ ω ≤ π ω ∨ j ≤ π ω) ∧ (τ ω ≤ j ∨ True) ** simp only [h, or_self_iff, and_true_iff] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω h : τ ω ≤ j ⊢ τ ω ≤ π ω ↔ τ ω ≤ π ω ∨ j ≤ π ω ** by_cases hj : j ≤ π ω ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω h : τ ω ≤ j hj : j ≤ π ω ⊢ τ ω ≤ π ω ↔ τ ω ≤ π ω ∨ j ≤ π ω ** simp only [hj, h.trans hj, or_self_iff] ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω h : τ ω ≤ j hj : ¬j ≤ π ω ⊢ τ ω ≤ π ω ↔ τ ω ≤ π ω ∨ j ≤ π ω ** simp only [hj, or_false_iff] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} ⊢ Measurable fun a => min (τ a) j ** exact (hτ.min_const j).measurable_of_le fun _ => min_le_right _ _ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁴ : TopologicalSpace ι inst✝³ : SecondCountableTopology ι inst✝² : OrderTopology ι inst✝¹ : MeasurableSpace ι inst✝ : BorelSpace ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} ⊢ Measurable fun a => min (π a) j ** exact (hπ.min_const j).measurable_of_le fun _ => min_le_right _ _ ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_eq_stopping_time ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π ⊢ MeasurableSet {ω | τ ω = π ω} ** rw [hτ.measurableSet] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π ⊢ ∀ (i : ι), MeasurableSet ({ω | τ ω = π ω} ∩ {ω | τ ω ≤ i}) ** intro j ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} ⊢ MeasurableSet ({ω | τ ω = π ω} ∩ {ω | τ ω ≤ j}) ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} ⊢ MeasurableSet ({ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j}) ** refine'
MeasurableSet.inter (MeasurableSet.inter _ (hτ.measurableSet_le j)) (hπ.measurableSet_le j) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} ⊢ MeasurableSet {ω | min (τ ω) j = min (π ω) j} ** apply measurableSet_eq_fun ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ⊢ {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω ⊢ ω ∈ {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} ↔ ω ∈ {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} ** simp only [Set.mem_inter_iff, Set.mem_setOf_eq] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω ⊢ τ ω = π ω ∧ τ ω ≤ j ↔ (min (τ ω) j = min (π ω) j ∧ τ ω ≤ j) ∧ π ω ≤ j ** refine' ⟨fun h => ⟨⟨_, h.2⟩, _⟩, fun h => ⟨_, h.1.2⟩⟩ ** case h.refine'_1 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω h : τ ω = π ω ∧ τ ω ≤ j ⊢ min (τ ω) j = min (π ω) j ** rw [h.1] ** case h.refine'_2 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω h : τ ω = π ω ∧ τ ω ≤ j ⊢ π ω ≤ j ** rw [← h.1] ** case h.refine'_2 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω h : τ ω = π ω ∧ τ ω ≤ j ⊢ τ ω ≤ j ** exact h.2 ** case h.refine'_3 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω h : (min (τ ω) j = min (π ω) j ∧ τ ω ≤ j) ∧ π ω ≤ j ⊢ τ ω = π ω ** cases' h with h' hσ_le ** case h.refine'_3.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω h' : min (τ ω) j = min (π ω) j ∧ τ ω ≤ j hσ_le : π ω ≤ j ⊢ τ ω = π ω ** cases' h' with h_eq hτ_le ** case h.refine'_3.intro.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι ω : Ω hσ_le : π ω ≤ j h_eq : min (τ ω) j = min (π ω) j hτ_le : τ ω ≤ j ⊢ τ ω = π ω ** rwa [min_eq_left hτ_le, min_eq_left hσ_le] at h_eq ** case hf Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} ⊢ Measurable fun x => min (τ x) j ** exact (hτ.min_const j).measurable_of_le fun _ => min_le_right _ _ ** case hg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁸ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝⁷ : AddGroup ι inst✝⁶ : TopologicalSpace ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : BorelSpace ι inst✝³ : OrderTopology ι inst✝² : MeasurableSingletonClass ι inst✝¹ : SecondCountableTopology ι inst✝ : MeasurableSub₂ ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π j : ι this : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} ⊢ Measurable fun x => min (π x) j ** exact (hπ.min_const j).measurable_of_le fun _ => min_le_right _ _ ** Qed
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MeasureTheory.stoppedProcess_eq_of_le ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι u : ι → Ω → β τ : Ω → ι i : ι ω : Ω h : i ≤ τ ω ⊢ stoppedProcess u τ i ω = u i ω ** simp [stoppedProcess, min_eq_left h] ** Qed
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MeasureTheory.stoppedProcess_eq_of_ge ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι u : ι → Ω → β τ : Ω → ι i : ι ω : Ω h : τ ω ≤ i ⊢ stoppedProcess u τ i ω = u (τ ω) ω ** simp [stoppedProcess, min_eq_right h] ** Qed
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MeasureTheory.stronglyMeasurable_stoppedValue_of_le ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁶ : LinearOrder ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : TopologicalSpace ι inst✝³ : OrderTopology ι inst✝² : SecondCountableTopology ι inst✝¹ : BorelSpace ι inst✝ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m h : ProgMeasurable f u hτ : IsStoppingTime f τ n : ι hτ_le : ∀ (ω : Ω), τ ω ≤ n ⊢ StronglyMeasurable (stoppedValue u τ) ** have : stoppedValue u τ =
(fun p : Set.Iic n × Ω => u (↑p.fst) p.snd) ∘ fun ω => (⟨τ ω, hτ_le ω⟩, ω) := by
ext1 ω; simp only [stoppedValue, Function.comp_apply, Subtype.coe_mk] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁶ : LinearOrder ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : TopologicalSpace ι inst✝³ : OrderTopology ι inst✝² : SecondCountableTopology ι inst✝¹ : BorelSpace ι inst✝ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m h : ProgMeasurable f u hτ : IsStoppingTime f τ n : ι hτ_le : ∀ (ω : Ω), τ ω ≤ n this : stoppedValue u τ = (fun p => u (↑p.1) p.2) ∘ fun ω => ({ val := τ ω, property := (_ : τ ω ≤ n) }, ω) ⊢ StronglyMeasurable (stoppedValue u τ) ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁶ : LinearOrder ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : TopologicalSpace ι inst✝³ : OrderTopology ι inst✝² : SecondCountableTopology ι inst✝¹ : BorelSpace ι inst✝ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m h : ProgMeasurable f u hτ : IsStoppingTime f τ n : ι hτ_le : ∀ (ω : Ω), τ ω ≤ n this : stoppedValue u τ = (fun p => u (↑p.1) p.2) ∘ fun ω => ({ val := τ ω, property := (_ : τ ω ≤ n) }, ω) ⊢ StronglyMeasurable ((fun p => u (↑p.1) p.2) ∘ fun ω => ({ val := τ ω, property := (_ : τ ω ≤ n) }, ω)) ** refine' StronglyMeasurable.comp_measurable (h n) _ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁶ : LinearOrder ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : TopologicalSpace ι inst✝³ : OrderTopology ι inst✝² : SecondCountableTopology ι inst✝¹ : BorelSpace ι inst✝ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m h : ProgMeasurable f u hτ : IsStoppingTime f τ n : ι hτ_le : ∀ (ω : Ω), τ ω ≤ n this : stoppedValue u τ = (fun p => u (↑p.1) p.2) ∘ fun ω => ({ val := τ ω, property := (_ : τ ω ≤ n) }, ω) ⊢ Measurable fun ω => ({ val := τ ω, property := (_ : τ ω ≤ n) }, ω) ** exact (hτ.measurable_of_le hτ_le).subtype_mk.prod_mk measurable_id ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁶ : LinearOrder ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : TopologicalSpace ι inst✝³ : OrderTopology ι inst✝² : SecondCountableTopology ι inst✝¹ : BorelSpace ι inst✝ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m h : ProgMeasurable f u hτ : IsStoppingTime f τ n : ι hτ_le : ∀ (ω : Ω), τ ω ≤ n ⊢ stoppedValue u τ = (fun p => u (↑p.1) p.2) ∘ fun ω => ({ val := τ ω, property := (_ : τ ω ≤ n) }, ω) ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁶ : LinearOrder ι inst✝⁵ : MeasurableSpace ι inst✝⁴ : TopologicalSpace ι inst✝³ : OrderTopology ι inst✝² : SecondCountableTopology ι inst✝¹ : BorelSpace ι inst✝ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m h : ProgMeasurable f u hτ : IsStoppingTime f τ n : ι hτ_le : ∀ (ω : Ω), τ ω ≤ n ω : Ω ⊢ stoppedValue u τ ω = ((fun p => u (↑p.1) p.2) ∘ fun ω => ({ val := τ ω, property := (_ : τ ω ≤ n) }, ω)) ω ** simp only [stoppedValue, Function.comp_apply, Subtype.coe_mk] ** Qed
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MeasureTheory.measurable_stoppedValue ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι inst✝⁸ : MeasurableSpace ι inst✝⁷ : TopologicalSpace ι inst✝⁶ : OrderTopology ι inst✝⁵ : SecondCountableTopology ι inst✝⁴ : BorelSpace ι inst✝³ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m inst✝² : MetrizableSpace β inst✝¹ : MeasurableSpace β inst✝ : BorelSpace β hf_prog : ProgMeasurable f u hτ : IsStoppingTime f τ ⊢ Measurable (stoppedValue u τ) ** have h_str_meas : ∀ i, StronglyMeasurable[f i] (stoppedValue u fun ω => min (τ ω) i) := fun i =>
stronglyMeasurable_stoppedValue_of_le hf_prog (hτ.min_const i) fun _ => min_le_right _ _ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι inst✝⁸ : MeasurableSpace ι inst✝⁷ : TopologicalSpace ι inst✝⁶ : OrderTopology ι inst✝⁵ : SecondCountableTopology ι inst✝⁴ : BorelSpace ι inst✝³ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m inst✝² : MetrizableSpace β inst✝¹ : MeasurableSpace β inst✝ : BorelSpace β hf_prog : ProgMeasurable f u hτ : IsStoppingTime f τ h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i) ⊢ Measurable (stoppedValue u τ) ** intro t ht i ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι inst✝⁸ : MeasurableSpace ι inst✝⁷ : TopologicalSpace ι inst✝⁶ : OrderTopology ι inst✝⁵ : SecondCountableTopology ι inst✝⁴ : BorelSpace ι inst✝³ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m inst✝² : MetrizableSpace β inst✝¹ : MeasurableSpace β inst✝ : BorelSpace β hf_prog : ProgMeasurable f u hτ : IsStoppingTime f τ h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i) t : Set β ht : MeasurableSet t i : ι ⊢ MeasurableSet (stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i}) ** suffices stoppedValue u τ ⁻¹' t ∩ {ω : Ω | τ ω ≤ i} =
(stoppedValue u fun ω => min (τ ω) i) ⁻¹' t ∩ {ω : Ω | τ ω ≤ i} by
rw [this]; exact ((h_str_meas i).measurable ht).inter (hτ.measurableSet_le i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι inst✝⁸ : MeasurableSpace ι inst✝⁷ : TopologicalSpace ι inst✝⁶ : OrderTopology ι inst✝⁵ : SecondCountableTopology ι inst✝⁴ : BorelSpace ι inst✝³ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m inst✝² : MetrizableSpace β inst✝¹ : MeasurableSpace β inst✝ : BorelSpace β hf_prog : ProgMeasurable f u hτ : IsStoppingTime f τ h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i) t : Set β ht : MeasurableSet t i : ι ⊢ stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i} = (stoppedValue u fun ω => min (τ ω) i) ⁻¹' t ∩ {ω | τ ω ≤ i} ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι inst✝⁸ : MeasurableSpace ι inst✝⁷ : TopologicalSpace ι inst✝⁶ : OrderTopology ι inst✝⁵ : SecondCountableTopology ι inst✝⁴ : BorelSpace ι inst✝³ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m inst✝² : MetrizableSpace β inst✝¹ : MeasurableSpace β inst✝ : BorelSpace β hf_prog : ProgMeasurable f u hτ : IsStoppingTime f τ h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i) t : Set β ht : MeasurableSet t i : ι ω : Ω ⊢ ω ∈ stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i} ↔ ω ∈ (stoppedValue u fun ω => min (τ ω) i) ⁻¹' t ∩ {ω | τ ω ≤ i} ** simp only [stoppedValue, Set.mem_inter_iff, Set.mem_preimage, Set.mem_setOf_eq,
and_congr_left_iff] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι inst✝⁸ : MeasurableSpace ι inst✝⁷ : TopologicalSpace ι inst✝⁶ : OrderTopology ι inst✝⁵ : SecondCountableTopology ι inst✝⁴ : BorelSpace ι inst✝³ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m inst✝² : MetrizableSpace β inst✝¹ : MeasurableSpace β inst✝ : BorelSpace β hf_prog : ProgMeasurable f u hτ : IsStoppingTime f τ h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i) t : Set β ht : MeasurableSet t i : ι ω : Ω ⊢ τ ω ≤ i → (u (τ ω) ω ∈ t ↔ u (min (τ ω) i) ω ∈ t) ** intro h ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι inst✝⁸ : MeasurableSpace ι inst✝⁷ : TopologicalSpace ι inst✝⁶ : OrderTopology ι inst✝⁵ : SecondCountableTopology ι inst✝⁴ : BorelSpace ι inst✝³ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m inst✝² : MetrizableSpace β inst✝¹ : MeasurableSpace β inst✝ : BorelSpace β hf_prog : ProgMeasurable f u hτ : IsStoppingTime f τ h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i) t : Set β ht : MeasurableSet t i : ι ω : Ω h : τ ω ≤ i ⊢ u (τ ω) ω ∈ t ↔ u (min (τ ω) i) ω ∈ t ** rw [min_eq_left h] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι inst✝⁸ : MeasurableSpace ι inst✝⁷ : TopologicalSpace ι inst✝⁶ : OrderTopology ι inst✝⁵ : SecondCountableTopology ι inst✝⁴ : BorelSpace ι inst✝³ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m inst✝² : MetrizableSpace β inst✝¹ : MeasurableSpace β inst✝ : BorelSpace β hf_prog : ProgMeasurable f u hτ : IsStoppingTime f τ h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i) t : Set β ht : MeasurableSet t i : ι this : stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i} = (stoppedValue u fun ω => min (τ ω) i) ⁻¹' t ∩ {ω | τ ω ≤ i} ⊢ MeasurableSet (stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i}) ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι inst✝⁸ : MeasurableSpace ι inst✝⁷ : TopologicalSpace ι inst✝⁶ : OrderTopology ι inst✝⁵ : SecondCountableTopology ι inst✝⁴ : BorelSpace ι inst✝³ : TopologicalSpace β u : ι → Ω → β τ : Ω → ι f : Filtration ι m inst✝² : MetrizableSpace β inst✝¹ : MeasurableSpace β inst✝ : BorelSpace β hf_prog : ProgMeasurable f u hτ : IsStoppingTime f τ h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i) t : Set β ht : MeasurableSet t i : ι this : stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ i} = (stoppedValue u fun ω => min (τ ω) i) ⁻¹' t ∩ {ω | τ ω ≤ i} ⊢ MeasurableSet ((stoppedValue u fun ω => min (τ ω) i) ⁻¹' t ∩ {ω | τ ω ≤ i}) ** exact ((h_str_meas i).measurable ht).inter (hτ.measurableSet_le i) ** Qed
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MeasureTheory.stoppedValue_eq_of_mem_finset ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s ⊢ stoppedValue u τ = ∑ i in s, Set.indicator {ω | τ ω = i} (u i) ** ext y ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω ⊢ stoppedValue u τ y = Finset.sum s (fun i => Set.indicator {ω | τ ω = i} (u i)) y ** rw [stoppedValue, Finset.sum_apply, Finset.sum_indicator_eq_sum_filter] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω ⊢ u (τ y) y = ∑ i in Finset.filter (fun i => y ∈ {ω | τ ω = i}) s, u i y ** suffices Finset.filter (fun i => y ∈ {ω : Ω | τ ω = i}) s = ({τ y} : Finset ι) by
rw [this, Finset.sum_singleton] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω ⊢ Finset.filter (fun i => y ∈ {ω | τ ω = i}) s = {τ y} ** ext1 ω ** case h.a Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω ω : ι ⊢ ω ∈ Finset.filter (fun i => y ∈ {ω | τ ω = i}) s ↔ ω ∈ {τ y} ** simp only [Set.mem_setOf_eq, Finset.mem_filter, Finset.mem_singleton] ** case h.a Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω ω : ι ⊢ ω ∈ s ∧ τ y = ω ↔ ω = τ y ** constructor <;> intro h ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω this : Finset.filter (fun i => y ∈ {ω | τ ω = i}) s = {τ y} ⊢ u (τ y) y = ∑ i in Finset.filter (fun i => y ∈ {ω | τ ω = i}) s, u i y ** rw [this, Finset.sum_singleton] ** case h.a.mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω ω : ι h : ω ∈ s ∧ τ y = ω ⊢ ω = τ y ** exact h.2.symm ** case h.a.mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω ω : ι h : ω = τ y ⊢ ω ∈ s ∧ τ y = ω ** refine' ⟨_, h.symm⟩ ** case h.a.mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω ω : ι h : ω = τ y ⊢ ω ∈ s ** rw [h] ** case h.a.mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝ : AddCommMonoid E s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s y : Ω ω : ι h : ω = τ y ⊢ τ y ∈ s ** exact hbdd y ** Qed
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MeasureTheory.stoppedProcess_eq'' ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝² : LinearOrder ι inst✝¹ : LocallyFiniteOrderBot ι inst✝ : AddCommMonoid E n : ι ⊢ stoppedProcess u τ n = Set.indicator {a | n ≤ τ a} (u n) + ∑ i in Finset.Iio n, Set.indicator {ω | τ ω = i} (u i) ** have h_mem : ∀ ω, τ ω < n → τ ω ∈ Finset.Iio n := fun ω h => Finset.mem_Iio.mpr h ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝² : LinearOrder ι inst✝¹ : LocallyFiniteOrderBot ι inst✝ : AddCommMonoid E n : ι h_mem : ∀ (ω : Ω), τ ω < n → τ ω ∈ Finset.Iio n ⊢ stoppedProcess u τ n = Set.indicator {a | n ≤ τ a} (u n) + ∑ i in Finset.Iio n, Set.indicator {ω | τ ω = i} (u i) ** rw [stoppedProcess_eq_of_mem_finset n h_mem] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝² : LinearOrder ι inst✝¹ : LocallyFiniteOrderBot ι inst✝ : AddCommMonoid E n : ι h_mem : ∀ (ω : Ω), τ ω < n → τ ω ∈ Finset.Iio n ⊢ Set.indicator {a | n ≤ τ a} (u n) + ∑ i in Finset.filter (fun x => x < n) (Finset.Iio n), Set.indicator {ω | τ ω = i} (u i) = Set.indicator {a | n ≤ τ a} (u n) + ∑ i in Finset.Iio n, Set.indicator {ω | τ ω = i} (u i) ** congr with i ** case e_a.e_s.a Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝² : LinearOrder ι inst✝¹ : LocallyFiniteOrderBot ι inst✝ : AddCommMonoid E n : ι h_mem : ∀ (ω : Ω), τ ω < n → τ ω ∈ Finset.Iio n i : ι ⊢ i ∈ Finset.filter (fun x => x < n) (Finset.Iio n) ↔ i ∈ Finset.Iio n ** simp ** Qed
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MeasureTheory.memℒp_stoppedValue_of_mem_finset ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝¹ : PartialOrder ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s ⊢ Memℒp (stoppedValue u τ) p ** rw [stoppedValue_eq_of_mem_finset hbdd] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝¹ : PartialOrder ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s ⊢ Memℒp (∑ i in s, Set.indicator {ω | τ ω = i} (u i)) p ** refine' memℒp_finset_sum' _ fun i _ => Memℒp.indicator _ (hu i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝¹ : PartialOrder ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s i : ι x✝ : i ∈ s ⊢ MeasurableSet {ω | τ ω = i} ** refine' ℱ.le i {a : Ω | τ a = i} (hτ.measurableSet_eq_of_countable_range _ i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝¹ : PartialOrder ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s i : ι x✝ : i ∈ s ⊢ Set.Countable (Set.range τ) ** refine' ((Finset.finite_toSet s).subset fun ω hω => _).countable ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝¹ : PartialOrder ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s i : ι x✝ : i ∈ s ω : ι hω : ω ∈ Set.range τ ⊢ ω ∈ ↑s ** obtain ⟨y, rfl⟩ := hω ** case intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝¹ : PartialOrder ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s i : ι x✝ : i ∈ s y : Ω ⊢ τ y ∈ ↑s ** exact hbdd y ** Qed
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MeasureTheory.integrable_stoppedValue_of_mem_finset ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝¹ : PartialOrder ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Integrable (u n) s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s ⊢ Integrable (stoppedValue u τ) ** simp_rw [← memℒp_one_iff_integrable] at hu ⊢ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝¹ : PartialOrder ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ s : Finset ι hbdd : ∀ (ω : Ω), τ ω ∈ s hu : ∀ (n : ι), Memℒp (u n) 1 ⊢ Memℒp (stoppedValue u τ) 1 ** exact memℒp_stoppedValue_of_mem_finset hτ hu hbdd ** Qed
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MeasureTheory.memℒp_stoppedProcess_of_mem_finset ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s ⊢ Memℒp (stoppedProcess u τ n) p ** rw [stoppedProcess_eq_of_mem_finset n hbdd] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s ⊢ Memℒp (Set.indicator {a | n ≤ τ a} (u n) + ∑ i in Finset.filter (fun x => x < n) s, Set.indicator {ω | τ ω = i} (u i)) p ** refine' Memℒp.add _ _ ** case refine'_1 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s ⊢ Memℒp (Set.indicator {a | n ≤ τ a} (u n)) p ** exact Memℒp.indicator (ℱ.le n {a : Ω | n ≤ τ a} (hτ.measurableSet_ge n)) (hu n) ** case refine'_2 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s ⊢ Memℒp (fun ω => ∑ i in Finset.filter (fun x => x < n) s, Set.indicator {a | τ a = i} (u i) ω) p ** refine' memℒp_finset_sum _ fun i _ => Memℒp.indicator _ (hu i) ** case refine'_2 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s i : ι x✝ : i ∈ Finset.filter (fun x => x < n) s ⊢ MeasurableSet {a | τ a = i} ** exact ℱ.le i {a : Ω | τ a = i} (hτ.measurableSet_eq i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s this : Memℒp (fun ω => ∑ i in Finset.filter (fun x => x < n) s, Set.indicator {a | τ a = i} (u i) ω) p ⊢ Memℒp (∑ i in Finset.filter (fun x => x < n) s, Set.indicator {ω | τ ω = i} (u i)) p ** convert this using 1 ** case h.e'_5 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s this : Memℒp (fun ω => ∑ i in Finset.filter (fun x => x < n) s, Set.indicator {a | τ a = i} (u i) ω) p ⊢ ∑ i in Finset.filter (fun x => x < n) s, Set.indicator {ω | τ ω = i} (u i) = fun ω => ∑ i in Finset.filter (fun x => x < n) s, Set.indicator {a | τ a = i} (u i) ω ** ext1 ω ** case h.e'_5.h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Memℒp (u n) p n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s this : Memℒp (fun ω => ∑ i in Finset.filter (fun x => x < n) s, Set.indicator {a | τ a = i} (u i) ω) p ω : Ω ⊢ Finset.sum (Finset.filter (fun x => x < n) s) (fun i => Set.indicator {ω | τ ω = i} (u i)) ω = ∑ i in Finset.filter (fun x => x < n) s, Set.indicator {a | τ a = i} (u i) ω ** simp only [Finset.sum_apply] ** Qed
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MeasureTheory.integrable_stoppedProcess_of_mem_finset ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ hu : ∀ (n : ι), Integrable (u n) n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s ⊢ Integrable (stoppedProcess u τ n) ** simp_rw [← memℒp_one_iff_integrable] at hu ⊢ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω μ : Measure Ω τ σ : Ω → ι E : Type u_4 p : ℝ≥0∞ u : ι → Ω → E inst✝⁴ : LinearOrder ι inst✝³ : TopologicalSpace ι inst✝² : OrderTopology ι inst✝¹ : FirstCountableTopology ι ℱ : Filtration ι m inst✝ : NormedAddCommGroup E hτ : IsStoppingTime ℱ τ n : ι s : Finset ι hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s hu : ∀ (n : ι), Memℒp (u n) 1 ⊢ Memℒp (stoppedProcess u τ n) 1 ** exact memℒp_stoppedProcess_of_mem_finset hτ hu n hbdd ** Qed
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MeasureTheory.stoppedValue_sub_eq_sum ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m u : ℕ → Ω → β τ π : Ω → ℕ inst✝ : AddCommGroup β hle : τ ≤ π ⊢ stoppedValue u π - stoppedValue u τ = fun ω => Finset.sum (Finset.Ico (τ ω) (π ω)) (fun i => u (i + 1) - u i) ω ** ext ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m u : ℕ → Ω → β τ π : Ω → ℕ inst✝ : AddCommGroup β hle : τ ≤ π ω : Ω ⊢ (stoppedValue u π - stoppedValue u τ) ω = Finset.sum (Finset.Ico (τ ω) (π ω)) (fun i => u (i + 1) - u i) ω ** rw [Finset.sum_Ico_eq_sub _ (hle ω), Finset.sum_range_sub, Finset.sum_range_sub] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m u : ℕ → Ω → β τ π : Ω → ℕ inst✝ : AddCommGroup β hle : τ ≤ π ω : Ω ⊢ (stoppedValue u π - stoppedValue u τ) ω = (u (π ω) - u 0 - (u (τ ω) - u 0)) ω ** simp [stoppedValue] ** Qed
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MeasureTheory.IsStoppingTime.piecewise_of_le ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s ⊢ IsStoppingTime 𝒢 (Set.piecewise s τ η) ** intro n ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι ⊢ MeasurableSet {ω | Set.piecewise s τ η ω ≤ n} ** have : {ω | s.piecewise τ η ω ≤ n} = s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} := by
ext1 ω
simp only [Set.piecewise, Set.mem_inter_iff, Set.mem_setOf_eq, and_congr_right_iff]
by_cases hx : ω ∈ s <;> simp [hx] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι this : {ω | Set.piecewise s τ η ω ≤ n} = s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} ⊢ MeasurableSet {ω | Set.piecewise s τ η ω ≤ n} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι this : {ω | Set.piecewise s τ η ω ≤ n} = s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n}) ** by_cases hin : i ≤ n ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι ⊢ {ω | Set.piecewise s τ η ω ≤ n} = s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι ω : Ω ⊢ ω ∈ {ω | Set.piecewise s τ η ω ≤ n} ↔ ω ∈ s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} ** simp only [Set.piecewise, Set.mem_inter_iff, Set.mem_setOf_eq, and_congr_right_iff] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι ω : Ω ⊢ (if ω ∈ s then τ ω else η ω) ≤ n ↔ ω ∈ s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} ** by_cases hx : ω ∈ s <;> simp [hx] ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι this : {ω | Set.piecewise s τ η ω ≤ n} = s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} hin : i ≤ n ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n}) ** have hs_n : MeasurableSet[𝒢 n] s := 𝒢.mono hin _ hs ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι this : {ω | Set.piecewise s τ η ω ≤ n} = s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} hin : i ≤ n hs_n : MeasurableSet s ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n}) ** exact (hs_n.inter (hτ_st n)).union (hs_n.compl.inter (hη_st n)) ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι this : {ω | Set.piecewise s τ η ω ≤ n} = s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} hin : ¬i ≤ n ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n}) ** have hτn : ∀ ω, ¬τ ω ≤ n := fun ω hτn => hin ((hτ ω).trans hτn) ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι this : {ω | Set.piecewise s τ η ω ≤ n} = s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} hin : ¬i ≤ n hτn : ∀ (ω : Ω), ¬τ ω ≤ n ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n}) ** have hηn : ∀ ω, ¬η ω ≤ n := fun ω hηn => hin ((hη ω).trans hηn) ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i j : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s hτ_st : IsStoppingTime 𝒢 τ hη_st : IsStoppingTime 𝒢 η hτ : ∀ (ω : Ω), i ≤ τ ω hη : ∀ (ω : Ω), i ≤ η ω hs : MeasurableSet s n : ι this : {ω | Set.piecewise s τ η ω ≤ n} = s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n} hin : ¬i ≤ n hτn : ∀ (ω : Ω), ¬τ ω ≤ n hηn : ∀ (ω : Ω), ¬η ω ≤ n ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ n} ∪ sᶜ ∩ {ω | η ω ≤ n}) ** simp [hτn, hηn, @MeasurableSet.empty _ _] ** Qed
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MeasureTheory.stoppedValue_piecewise_const ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i✝ j✝ : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s ι' : Type u_4 i j : ι' f : ι' → Ω → ℝ ⊢ stoppedValue f (Set.piecewise s (fun x => i) fun x => j) = Set.piecewise s (f i) (f j) ** ext ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i✝ j✝ : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s ι' : Type u_4 i j : ι' f : ι' → Ω → ℝ ω : Ω ⊢ stoppedValue f (Set.piecewise s (fun x => i) fun x => j) ω = Set.piecewise s (f i) (f j) ω ** rw [stoppedValue] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i✝ j✝ : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s ι' : Type u_4 i j : ι' f : ι' → Ω → ℝ ω : Ω ⊢ f (Set.piecewise s (fun x => i) (fun x => j) ω) ω = Set.piecewise s (f i) (f j) ω ** by_cases hx : ω ∈ s <;> simp [hx] ** Qed
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MeasureTheory.stoppedValue_piecewise_const' ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i✝ j✝ : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s ι' : Type u_4 i j : ι' f : ι' → Ω → ℝ ⊢ stoppedValue f (Set.piecewise s (fun x => i) fun x => j) = Set.indicator s (f i) + Set.indicator sᶜ (f j) ** ext ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i✝ j✝ : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s ι' : Type u_4 i j : ι' f : ι' → Ω → ℝ ω : Ω ⊢ stoppedValue f (Set.piecewise s (fun x => i) fun x => j) ω = (Set.indicator s (f i) + Set.indicator sᶜ (f j)) ω ** rw [stoppedValue] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι 𝒢 : Filtration ι m τ η : Ω → ι i✝ j✝ : ι s : Set Ω inst✝ : DecidablePred fun x => x ∈ s ι' : Type u_4 i j : ι' f : ι' → Ω → ℝ ω : Ω ⊢ f (Set.piecewise s (fun x => i) (fun x => j) ω) ω = (Set.indicator s (f i) + Set.indicator sᶜ (f j)) ω ** by_cases hx : ω ∈ s <;> simp [hx] ** Qed
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MeasureTheory.condexp_stopping_time_ae_eq_restrict_eq_of_countable_range ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f : Ω → E inst✝¹ : SigmaFiniteFiltration μ ℱ hτ : IsStoppingTime ℱ τ h_countable : Set.Countable (Set.range τ) inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace hτ ≤ m)) i : ι ⊢ μ[f|IsStoppingTime.measurableSpace hτ] =ᵐ[Measure.restrict μ {x | τ x = i}] μ[f|↑ℱ i] ** refine' condexp_ae_eq_restrict_of_measurableSpace_eq_on
(hτ.measurableSpace_le_of_countable_range h_countable) (ℱ.le i)
(hτ.measurableSet_eq_of_countable_range' h_countable i) fun t => _ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f : Ω → E inst✝¹ : SigmaFiniteFiltration μ ℱ hτ : IsStoppingTime ℱ τ h_countable : Set.Countable (Set.range τ) inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace hτ ≤ m)) i : ι t : Set Ω ⊢ MeasurableSet ({x | τ x = i} ∩ t) ↔ MeasurableSet ({x | τ x = i} ∩ t) ** rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff] ** Qed
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MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le_const ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f : Ω → E inst✝¹ : IsCountablyGenerated atTop hτ : IsStoppingTime ℱ τ i : ι inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime ℱ fun ω => min (τ ω) i) ≤ m)) ⊢ μ[f|IsStoppingTime.measurableSpace (_ : IsStoppingTime ℱ fun ω => min (τ ω) i)] =ᵐ[Measure.restrict μ {x | τ x ≤ i}] μ[f|IsStoppingTime.measurableSpace hτ] ** have : SigmaFinite (μ.trim hτ.measurableSpace_le) :=
haveI h_le : (hτ.min_const i).measurableSpace ≤ hτ.measurableSpace := by
rw [IsStoppingTime.measurableSpace_min_const]
exact inf_le_left
sigmaFiniteTrim_mono _ h_le ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f : Ω → E inst✝¹ : IsCountablyGenerated atTop hτ : IsStoppingTime ℱ τ i : ι inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime ℱ fun ω => min (τ ω) i) ≤ m)) this : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace hτ ≤ m)) ⊢ μ[f|IsStoppingTime.measurableSpace (_ : IsStoppingTime ℱ fun ω => min (τ ω) i)] =ᵐ[Measure.restrict μ {x | τ x ≤ i}] μ[f|IsStoppingTime.measurableSpace hτ] ** refine' (condexp_ae_eq_restrict_of_measurableSpace_eq_on hτ.measurableSpace_le
(hτ.min_const i).measurableSpace_le (hτ.measurableSet_le' i) fun t => _).symm ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f : Ω → E inst✝¹ : IsCountablyGenerated atTop hτ : IsStoppingTime ℱ τ i : ι inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime ℱ fun ω => min (τ ω) i) ≤ m)) this : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace hτ ≤ m)) t : Set Ω ⊢ MeasurableSet ({ω | τ ω ≤ i} ∩ t) ↔ MeasurableSet ({ω | τ ω ≤ i} ∩ t) ** rw [Set.inter_comm _ t, hτ.measurableSet_inter_le_const_iff] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f : Ω → E inst✝¹ : IsCountablyGenerated atTop hτ : IsStoppingTime ℱ τ i : ι inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime ℱ fun ω => min (τ ω) i) ≤ m)) ⊢ IsStoppingTime.measurableSpace (_ : IsStoppingTime ℱ fun ω => min (τ ω) i) ≤ IsStoppingTime.measurableSpace hτ ** rw [IsStoppingTime.measurableSpace_min_const] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f : Ω → E inst✝¹ : IsCountablyGenerated atTop hτ : IsStoppingTime ℱ τ i : ι inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime ℱ fun ω => min (τ ω) i) ≤ m)) ⊢ IsStoppingTime.measurableSpace ?hτ ⊓ ↑ℱ i ≤ IsStoppingTime.measurableSpace hτ case hτ Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁵ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f : Ω → E inst✝¹ : IsCountablyGenerated atTop hτ : IsStoppingTime ℱ τ i : ι inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace (_ : IsStoppingTime ℱ fun ω => min (τ ω) i) ≤ m)) ⊢ IsStoppingTime ℱ fun ω => τ ω ** exact inf_le_left ** Qed
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MeasureTheory.condexp_stopping_time_ae_eq_restrict_eq ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : CompleteSpace E f : Ω → E inst✝⁵ : IsCountablyGenerated atTop inst✝⁴ : TopologicalSpace ι inst✝³ : OrderTopology ι inst✝² : FirstCountableTopology ι inst✝¹ : SigmaFiniteFiltration μ ℱ hτ : IsStoppingTime ℱ τ inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace hτ ≤ m)) i : ι ⊢ μ[f|IsStoppingTime.measurableSpace hτ] =ᵐ[Measure.restrict μ {x | τ x = i}] μ[f|↑ℱ i] ** refine' condexp_ae_eq_restrict_of_measurableSpace_eq_on hτ.measurableSpace_le (ℱ.le i)
(hτ.measurableSet_eq' i) fun t => _ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝⁹ : LinearOrder ι μ : Measure Ω ℱ : Filtration ι m τ σ : Ω → ι E : Type u_4 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : CompleteSpace E f : Ω → E inst✝⁵ : IsCountablyGenerated atTop inst✝⁴ : TopologicalSpace ι inst✝³ : OrderTopology ι inst✝² : FirstCountableTopology ι inst✝¹ : SigmaFiniteFiltration μ ℱ hτ : IsStoppingTime ℱ τ inst✝ : SigmaFinite (Measure.trim μ (_ : IsStoppingTime.measurableSpace hτ ≤ m)) i : ι t : Set Ω ⊢ MeasurableSet ({x | τ x = i} ∩ t) ↔ MeasurableSet ({x | τ x = i} ∩ t) ** rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff] ** Qed
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PMF.binomial_apply_zero ** p : ℝ≥0∞ h : p ≤ 1 n : ℕ ⊢ ↑(binomial p h n) 0 = (1 - p) ^ n ** simp [binomial_apply] ** Qed
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PMF.binomial_apply_self ** p : ℝ≥0∞ h : p ≤ 1 n : ℕ ⊢ ↑(binomial p h n) ↑n = p ^ n ** simp [binomial_apply, Nat.mod_eq_of_lt] ** Qed
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MeasureTheory.AEStronglyMeasurable.truncation ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ hf : AEStronglyMeasurable f μ A : ℝ ⊢ AEStronglyMeasurable (ProbabilityTheory.truncation f A) μ ** apply AEStronglyMeasurable.comp_aemeasurable _ hf.aemeasurable ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ hf : AEStronglyMeasurable f μ A : ℝ ⊢ AEStronglyMeasurable (indicator (Set.Ioc (-A) A) id) (Measure.map f μ) ** exact (stronglyMeasurable_id.indicator measurableSet_Ioc).aestronglyMeasurable ** Qed
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ProbabilityTheory.abs_truncation_le_bound ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α ⊢ |truncation f A x| ≤ |A| ** simp only [truncation, Set.indicator, Set.mem_Icc, id.def, Function.comp_apply] ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α ⊢ |if f x ∈ Set.Ioc (-A) A then f x else 0| ≤ |A| ** split_ifs with h ** case pos α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α h : f x ∈ Set.Ioc (-A) A ⊢ |f x| ≤ |A| ** exact abs_le_abs h.2 (neg_le.2 h.1.le) ** case neg α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α h : ¬f x ∈ Set.Ioc (-A) A ⊢ |0| ≤ |A| ** simp [abs_nonneg] ** Qed
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ProbabilityTheory.truncation_zero ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ ⊢ truncation f 0 = 0 ** simp [truncation] ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ ⊢ (fun x => 0) ∘ f = 0 ** rfl ** Qed
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ProbabilityTheory.abs_truncation_le_abs_self ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α ⊢ |truncation f A x| ≤ |f x| ** simp only [truncation, indicator, Set.mem_Icc, id.def, Function.comp_apply] ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α ⊢ |if f x ∈ Set.Ioc (-A) A then f x else 0| ≤ |f x| ** split_ifs ** case pos α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α h✝ : f x ∈ Set.Ioc (-A) A ⊢ |f x| ≤ |f x| ** exact le_rfl ** case neg α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α h✝ : ¬f x ∈ Set.Ioc (-A) A ⊢ |0| ≤ |f x| ** simp [abs_nonneg] ** Qed
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ProbabilityTheory.truncation_eq_self ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α h : |f x| < A ⊢ truncation f A x = f x ** simp only [truncation, indicator, Set.mem_Icc, id.def, Function.comp_apply, ite_eq_left_iff] ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α h : |f x| < A ⊢ ¬f x ∈ Set.Ioc (-A) A → 0 = f x ** intro H ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α h : |f x| < A H : ¬f x ∈ Set.Ioc (-A) A ⊢ 0 = f x ** apply H.elim ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ x : α h : |f x| < A H : ¬f x ∈ Set.Ioc (-A) A ⊢ f x ∈ Set.Ioc (-A) A ** simp [(abs_lt.1 h).1, (abs_lt.1 h).2.le] ** Qed
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ProbabilityTheory.truncation_eq_of_nonneg ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ h : ∀ (x : α), 0 ≤ f x ⊢ truncation f A = indicator (Set.Ioc 0 A) id ∘ f ** ext x ** case h α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ h : ∀ (x : α), 0 ≤ f x x : α ⊢ truncation f A x = (indicator (Set.Ioc 0 A) id ∘ f) x ** rcases (h x).lt_or_eq with (hx | hx) ** case h.inl α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ h : ∀ (x : α), 0 ≤ f x x : α hx : 0 < f x ⊢ truncation f A x = (indicator (Set.Ioc 0 A) id ∘ f) x ** simp only [truncation, indicator, hx, Set.mem_Ioc, id.def, Function.comp_apply, true_and_iff] ** case h.inl α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ h : ∀ (x : α), 0 ≤ f x x : α hx : 0 < f x ⊢ (if -A < f x ∧ f x ≤ A then f x else 0) = if f x ≤ A then f x else 0 ** by_cases h'x : f x ≤ A ** case pos α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ h : ∀ (x : α), 0 ≤ f x x : α hx : 0 < f x h'x : f x ≤ A ⊢ (if -A < f x ∧ f x ≤ A then f x else 0) = if f x ≤ A then f x else 0 ** have : -A < f x := by linarith [h x] ** case pos α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ h : ∀ (x : α), 0 ≤ f x x : α hx : 0 < f x h'x : f x ≤ A this : -A < f x ⊢ (if -A < f x ∧ f x ≤ A then f x else 0) = if f x ≤ A then f x else 0 ** simp only [this, true_and_iff] ** α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ h : ∀ (x : α), 0 ≤ f x x : α hx : 0 < f x h'x : f x ≤ A ⊢ -A < f x ** linarith [h x] ** case neg α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ h : ∀ (x : α), 0 ≤ f x x : α hx : 0 < f x h'x : ¬f x ≤ A ⊢ (if -A < f x ∧ f x ≤ A then f x else 0) = if f x ≤ A then f x else 0 ** simp only [h'x, and_false_iff] ** case h.inr α : Type u_1 m : MeasurableSpace α μ : Measure α f✝ f : α → ℝ A : ℝ h : ∀ (x : α), 0 ≤ f x x : α hx : 0 = f x ⊢ truncation f A x = (indicator (Set.Ioc 0 A) id ∘ f) x ** simp only [truncation, indicator, hx, id.def, Function.comp_apply, ite_self] ** Qed
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MeasureTheory.AEStronglyMeasurable.integrable_truncation ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ inst✝ : IsFiniteMeasure μ hf : AEStronglyMeasurable f μ A : ℝ ⊢ Integrable (truncation f A) ** rw [← memℒp_one_iff_integrable] ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ inst✝ : IsFiniteMeasure μ hf : AEStronglyMeasurable f μ A : ℝ ⊢ Memℒp (truncation f A) 1 ** exact hf.memℒp_truncation ** Qed
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ProbabilityTheory.moment_truncation_eq_intervalIntegral ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ hf : AEStronglyMeasurable f μ A : ℝ hA : 0 ≤ A n : ℕ hn : n ≠ 0 ⊢ ∫ (x : α), truncation f A x ^ n ∂μ = ∫ (y : ℝ) in -A..A, y ^ n ∂Measure.map f μ ** have M : MeasurableSet (Set.Ioc (-A) A) := measurableSet_Ioc ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ hf : AEStronglyMeasurable f μ A : ℝ hA : 0 ≤ A n : ℕ hn : n ≠ 0 M : MeasurableSet (Set.Ioc (-A) A) ⊢ ∫ (x : α), truncation f A x ^ n ∂μ = ∫ (y : ℝ) in -A..A, y ^ n ∂Measure.map f μ ** change ∫ x, (fun z => indicator (Set.Ioc (-A) A) id z ^ n) (f x) ∂μ = _ ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ hf : AEStronglyMeasurable f μ A : ℝ hA : 0 ≤ A n : ℕ hn : n ≠ 0 M : MeasurableSet (Set.Ioc (-A) A) ⊢ ∫ (x : α), (fun z => indicator (Set.Ioc (-A) A) id z ^ n) (f x) ∂μ = ∫ (y : ℝ) in -A..A, y ^ n ∂Measure.map f μ ** rw [← integral_map (f := fun z => _ ^ n) hf.aemeasurable, intervalIntegral.integral_of_le,
← integral_indicator M] ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ hf : AEStronglyMeasurable f μ A : ℝ hA : 0 ≤ A n : ℕ hn : n ≠ 0 M : MeasurableSet (Set.Ioc (-A) A) ⊢ ∫ (y : ℝ), indicator (Set.Ioc (-A) A) id y ^ n ∂Measure.map f μ = ∫ (x : ℝ), indicator (Set.Ioc (-A) A) (fun x => x ^ n) x ∂Measure.map f μ ** simp only [indicator, zero_pow' _ hn, id.def, ite_pow] ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ hf : AEStronglyMeasurable f μ A : ℝ hA : 0 ≤ A n : ℕ hn : n ≠ 0 M : MeasurableSet (Set.Ioc (-A) A) ⊢ -A ≤ A ** linarith ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ hf : AEStronglyMeasurable f μ A : ℝ hA : 0 ≤ A n : ℕ hn : n ≠ 0 M : MeasurableSet (Set.Ioc (-A) A) ⊢ AEStronglyMeasurable (fun z => indicator (Set.Ioc (-A) A) id z ^ n) (Measure.map f μ) ** exact ((measurable_id.indicator M).pow_const n).aestronglyMeasurable ** Qed
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ProbabilityTheory.integral_truncation_eq_intervalIntegral_of_nonneg ** α : Type u_1 m : MeasurableSpace α μ : Measure α f : α → ℝ hf : AEStronglyMeasurable f μ A : ℝ h'f : 0 ≤ f ⊢ ∫ (x : α), truncation f A x ∂μ = ∫ (y : ℝ) in 0 ..A, y ∂Measure.map f μ ** simpa using moment_truncation_eq_intervalIntegral_of_nonneg hf one_ne_zero h'f ** Qed
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ProbabilityTheory.strong_law_ae_real ** Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ⊢ ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, X i ω) / ↑n) atTop (𝓝 (∫ (a : Ω), X 0 a)) ** let pos : ℝ → ℝ := fun x => max x 0 ** Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) pos : ℝ → ℝ := fun x => max x 0 ⊢ ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, X i ω) / ↑n) atTop (𝓝 (∫ (a : Ω), X 0 a)) ** let neg : ℝ → ℝ := fun x => max (-x) 0 ** Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) pos : ℝ → ℝ := fun x => max x 0 neg : ℝ → ℝ := fun x => max (-x) 0 ⊢ ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, X i ω) / ↑n) atTop (𝓝 (∫ (a : Ω), X 0 a)) ** have posm : Measurable pos := measurable_id'.max measurable_const ** Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) pos : ℝ → ℝ := fun x => max x 0 neg : ℝ → ℝ := fun x => max (-x) 0 posm : Measurable pos ⊢ ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, X i ω) / ↑n) atTop (𝓝 (∫ (a : Ω), X 0 a)) ** have negm : Measurable neg := measurable_id'.neg.max measurable_const ** Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) pos : ℝ → ℝ := fun x => max x 0 neg : ℝ → ℝ := fun x => max (-x) 0 posm : Measurable pos negm : Measurable neg ⊢ ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, X i ω) / ↑n) atTop (𝓝 (∫ (a : Ω), X 0 a)) ** have A: ∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i in range n, (pos ∘ X i) ω) / n) atTop (𝓝 𝔼[pos ∘ X 0]) :=
strong_law_aux7 _ hint.pos_part (fun i j hij => (hindep hij).comp posm posm)
(fun i => (hident i).comp posm) fun i ω => le_max_right _ _ ** Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) pos : ℝ → ℝ := fun x => max x 0 neg : ℝ → ℝ := fun x => max (-x) 0 posm : Measurable pos negm : Measurable neg A : ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, (pos ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (pos ∘ X 0) a)) ⊢ ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, X i ω) / ↑n) atTop (𝓝 (∫ (a : Ω), X 0 a)) ** have B: ∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i in range n, (neg ∘ X i) ω) / n) atTop (𝓝 𝔼[neg ∘ X 0]) :=
strong_law_aux7 _ hint.neg_part (fun i j hij => (hindep hij).comp negm negm)
(fun i => (hident i).comp negm) fun i ω => le_max_right _ _ ** Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) pos : ℝ → ℝ := fun x => max x 0 neg : ℝ → ℝ := fun x => max (-x) 0 posm : Measurable pos negm : Measurable neg A : ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, (pos ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (pos ∘ X 0) a)) B : ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, (neg ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (neg ∘ X 0) a)) ⊢ ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, X i ω) / ↑n) atTop (𝓝 (∫ (a : Ω), X 0 a)) ** filter_upwards [A, B] with ω hωpos hωneg ** case h Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) pos : ℝ → ℝ := fun x => max x 0 neg : ℝ → ℝ := fun x => max (-x) 0 posm : Measurable pos negm : Measurable neg A : ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, (pos ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (pos ∘ X 0) a)) B : ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, (neg ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (neg ∘ X 0) a)) ω : Ω hωpos : Tendsto (fun n => (∑ i in range n, (pos ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (pos ∘ X 0) a)) hωneg : Tendsto (fun n => (∑ i in range n, (neg ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (neg ∘ X 0) a)) ⊢ Tendsto (fun n => (∑ i in range n, X i ω) / ↑n) atTop (𝓝 (∫ (a : Ω), X 0 a)) ** convert hωpos.sub hωneg using 1 ** case h.e'_3 Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) pos : ℝ → ℝ := fun x => max x 0 neg : ℝ → ℝ := fun x => max (-x) 0 posm : Measurable pos negm : Measurable neg A : ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, (pos ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (pos ∘ X 0) a)) B : ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, (neg ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (neg ∘ X 0) a)) ω : Ω hωpos : Tendsto (fun n => (∑ i in range n, (pos ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (pos ∘ X 0) a)) hωneg : Tendsto (fun n => (∑ i in range n, (neg ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (neg ∘ X 0) a)) ⊢ (fun n => (∑ i in range n, X i ω) / ↑n) = fun x => (∑ i in range x, (pos ∘ X i) ω) / ↑x - (∑ i in range x, (neg ∘ X i) ω) / ↑x ** simp only [← sub_div, ← sum_sub_distrib, max_zero_sub_max_neg_zero_eq_self, Function.comp_apply] ** case h.e'_5 Ω : Type u_1 inst✝¹ : MeasureSpace Ω inst✝ : IsProbabilityMeasure ℙ X : ℕ → Ω → ℝ hint : Integrable (X 0) hindep : Pairwise fun i j => IndepFun (X i) (X j) hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) pos : ℝ → ℝ := fun x => max x 0 neg : ℝ → ℝ := fun x => max (-x) 0 posm : Measurable pos negm : Measurable neg A : ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, (pos ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (pos ∘ X 0) a)) B : ∀ᵐ (ω : Ω), Tendsto (fun n => (∑ i in range n, (neg ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (neg ∘ X 0) a)) ω : Ω hωpos : Tendsto (fun n => (∑ i in range n, (pos ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (pos ∘ X 0) a)) hωneg : Tendsto (fun n => (∑ i in range n, (neg ∘ X i) ω) / ↑n) atTop (𝓝 (∫ (a : Ω), (neg ∘ X 0) a)) ⊢ 𝓝 (∫ (a : Ω), X 0 a) = 𝓝 ((∫ (a : Ω), (pos ∘ X 0) a) - ∫ (a : Ω), (neg ∘ X 0) a) ** simp only [← integral_sub hint.pos_part hint.neg_part, max_zero_sub_max_neg_zero_eq_self,
Function.comp_apply] ** Qed
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ProbabilityTheory.kernel.compProdFun_empty ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } a : α ⊢ compProdFun κ η a ∅ = 0 ** simp only [compProdFun, Set.mem_empty_iff_false, Set.setOf_false, measure_empty,
MeasureTheory.lintegral_const, zero_mul] ** Qed
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ProbabilityTheory.kernel.compProdFun_iUnion ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) ⊢ compProdFun κ η a (⋃ i, f i) = ∑' (i : ℕ), compProdFun κ η a (f i) ** have h_Union :
(fun b => η (a, b) {c : γ | (b, c) ∈ ⋃ i, f i}) = fun b =>
η (a, b) (⋃ i, {c : γ | (b, c) ∈ f i}) := by
ext1 b
congr with c
simp only [Set.mem_iUnion, Set.iSup_eq_iUnion, Set.mem_setOf_eq] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) ⊢ compProdFun κ η a (⋃ i, f i) = ∑' (i : ℕ), compProdFun κ η a (f i) ** rw [compProdFun, h_Union] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) h_tsum : (fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i})) = fun b => ∑' (i : ℕ), ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} ⊢ ∫⁻ (b : β), ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) ∂↑κ a = ∑' (i : ℕ), compProdFun κ η a (f i) ** rw [h_tsum, lintegral_tsum] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) ⊢ (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) ** ext1 b ** case h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) b : β ⊢ ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i} = ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) ** congr with c ** case h.e_a.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) b : β c : γ ⊢ c ∈ {c | (b, c) ∈ ⋃ i, f i} ↔ c ∈ ⋃ i, {c | (b, c) ∈ f i} ** simp only [Set.mem_iUnion, Set.iSup_eq_iUnion, Set.mem_setOf_eq] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) ⊢ (fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i})) = fun b => ∑' (i : ℕ), ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} ** ext1 b ** case h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β ⊢ ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) = ∑' (i : ℕ), ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} ** rw [measure_iUnion] ** case h.hn α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β ⊢ Pairwise (Disjoint on fun i => {c | (b, c) ∈ f i}) ** intro i j hij s hsi hsj c hcs ** case h.hn α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β i j : ℕ hij : i ≠ j s : Set γ hsi : s ≤ (fun i => {c | (b, c) ∈ f i}) i hsj : s ≤ (fun i => {c | (b, c) ∈ f i}) j c : γ hcs : c ∈ s ⊢ c ∈ ⊥ ** have hbci : {(b, c)} ⊆ f i := by rw [Set.singleton_subset_iff]; exact hsi hcs ** case h.hn α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β i j : ℕ hij : i ≠ j s : Set γ hsi : s ≤ (fun i => {c | (b, c) ∈ f i}) i hsj : s ≤ (fun i => {c | (b, c) ∈ f i}) j c : γ hcs : c ∈ s hbci : {(b, c)} ⊆ f i ⊢ c ∈ ⊥ ** have hbcj : {(b, c)} ⊆ f j := by rw [Set.singleton_subset_iff]; exact hsj hcs ** case h.hn α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β i j : ℕ hij : i ≠ j s : Set γ hsi : s ≤ (fun i => {c | (b, c) ∈ f i}) i hsj : s ≤ (fun i => {c | (b, c) ∈ f i}) j c : γ hcs : c ∈ s hbci : {(b, c)} ⊆ f i hbcj : {(b, c)} ⊆ f j ⊢ c ∈ ⊥ ** simpa only [Set.bot_eq_empty, Set.le_eq_subset, Set.singleton_subset_iff,
Set.mem_empty_iff_false] using hf_disj hij hbci hbcj ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β i j : ℕ hij : i ≠ j s : Set γ hsi : s ≤ (fun i => {c | (b, c) ∈ f i}) i hsj : s ≤ (fun i => {c | (b, c) ∈ f i}) j c : γ hcs : c ∈ s ⊢ {(b, c)} ⊆ f i ** rw [Set.singleton_subset_iff] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β i j : ℕ hij : i ≠ j s : Set γ hsi : s ≤ (fun i => {c | (b, c) ∈ f i}) i hsj : s ≤ (fun i => {c | (b, c) ∈ f i}) j c : γ hcs : c ∈ s ⊢ (b, c) ∈ f i ** exact hsi hcs ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β i j : ℕ hij : i ≠ j s : Set γ hsi : s ≤ (fun i => {c | (b, c) ∈ f i}) i hsj : s ≤ (fun i => {c | (b, c) ∈ f i}) j c : γ hcs : c ∈ s hbci : {(b, c)} ⊆ f i ⊢ {(b, c)} ⊆ f j ** rw [Set.singleton_subset_iff] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β i j : ℕ hij : i ≠ j s : Set γ hsi : s ≤ (fun i => {c | (b, c) ∈ f i}) i hsj : s ≤ (fun i => {c | (b, c) ∈ f i}) j c : γ hcs : c ∈ s hbci : {(b, c)} ⊆ f i ⊢ (b, c) ∈ f j ** exact hsj hcs ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) b : β ⊢ ∀ (i : ℕ), MeasurableSet {c | (b, c) ∈ f i} ** exact fun i => (@measurable_prod_mk_left β γ _ _ b) (hf_meas i) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) h_tsum : (fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i})) = fun b => ∑' (i : ℕ), ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} ⊢ ∑' (i : ℕ), ∫⁻ (a_1 : β), ↑↑(↑η (a, a_1)) {c | (a_1, c) ∈ f i} ∂↑κ a = ∑' (i : ℕ), compProdFun κ η a (f i) ** rfl ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) h_tsum : (fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i})) = fun b => ∑' (i : ℕ), ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} ⊢ ∀ (i : ℕ), AEMeasurable fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} ** intro i ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) h_tsum : (fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i})) = fun b => ∑' (i : ℕ), ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} i : ℕ ⊢ AEMeasurable fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} ** have hm : MeasurableSet {p : (α × β) × γ | (p.1.2, p.2) ∈ f i} :=
measurable_fst.snd.prod_mk measurable_snd (hf_meas i) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : ℕ → Set (β × γ) hf_meas : ∀ (i : ℕ), MeasurableSet (f i) hf_disj : Pairwise (Disjoint on f) h_Union : (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i}) h_tsum : (fun b => ↑↑(↑η (a, b)) (⋃ i, {c | (b, c) ∈ f i})) = fun b => ∑' (i : ℕ), ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} i : ℕ hm : MeasurableSet {p | (p.1.2, p.2) ∈ f i} ⊢ AEMeasurable fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ f i} ** exact ((measurable_kernel_prod_mk_left hm).comp measurable_prod_mk_left).aemeasurable ** Qed
| |
ProbabilityTheory.kernel.compProdFun_tsum_right ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α hs : MeasurableSet s ⊢ compProdFun κ η a s = ∑' (n : ℕ), compProdFun κ (seq η n) a s ** simp_rw [compProdFun, (measure_sum_seq η _).symm] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α hs : MeasurableSet s ⊢ ∫⁻ (b : β), ↑↑(Measure.sum fun n => ↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a = ∑' (n : ℕ), ∫⁻ (b : β), ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a ** have :
∫⁻ b, Measure.sum (fun n => seq η n (a, b)) {c : γ | (b, c) ∈ s} ∂κ a =
∫⁻ b, ∑' n, seq η n (a, b) {c : γ | (b, c) ∈ s} ∂κ a := by
congr
ext1 b
rw [Measure.sum_apply]
exact measurable_prod_mk_left hs ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α hs : MeasurableSet s this : ∫⁻ (b : β), ↑↑(Measure.sum fun n => ↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a = ∫⁻ (b : β), ∑' (n : ℕ), ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a ⊢ ∫⁻ (b : β), ↑↑(Measure.sum fun n => ↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a = ∑' (n : ℕ), ∫⁻ (b : β), ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a ** rw [this, lintegral_tsum] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α hs : MeasurableSet s this : ∫⁻ (b : β), ↑↑(Measure.sum fun n => ↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a = ∫⁻ (b : β), ∑' (n : ℕ), ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a ⊢ ∀ (i : ℕ), AEMeasurable fun b => ↑↑(↑(seq η i) (a, b)) {c | (b, c) ∈ s} ** exact fun n => ((measurable_kernel_prod_mk_left (κ := (seq η n))
((measurable_fst.snd.prod_mk measurable_snd) hs)).comp measurable_prod_mk_left).aemeasurable ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α hs : MeasurableSet s ⊢ ∫⁻ (b : β), ↑↑(Measure.sum fun n => ↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a = ∫⁻ (b : β), ∑' (n : ℕ), ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a ** congr ** case e_f α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α hs : MeasurableSet s ⊢ (fun b => ↑↑(Measure.sum fun n => ↑(seq η n) (a, b)) {c | (b, c) ∈ s}) = fun b => ∑' (n : ℕ), ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s} ** ext1 b ** case e_f.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α hs : MeasurableSet s b : β ⊢ ↑↑(Measure.sum fun n => ↑(seq η n) (a, b)) {c | (b, c) ∈ s} = ∑' (n : ℕ), ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s} ** rw [Measure.sum_apply] ** case e_f.h.hs α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α hs : MeasurableSet s b : β ⊢ MeasurableSet {c | (b, c) ∈ s} ** exact measurable_prod_mk_left hs ** Qed
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ProbabilityTheory.kernel.compProdFun_tsum_left ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel κ a : α s : Set (β × γ) ⊢ compProdFun κ η a s = ∑' (n : ℕ), compProdFun (seq κ n) η a s ** simp_rw [compProdFun, (measure_sum_seq κ _).symm, lintegral_sum_measure] ** Qed
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ProbabilityTheory.kernel.measurable_compProdFun_of_finite ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η hs : MeasurableSet s ⊢ Measurable fun a => compProdFun κ η a s ** simp only [compProdFun] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η hs : MeasurableSet s ⊢ Measurable fun a => ∫⁻ (b : β), ↑↑(↑η (a, b)) {c | (b, c) ∈ s} ∂↑κ a ** have h_meas : Measurable (Function.uncurry fun a b => η (a, b) {c : γ | (b, c) ∈ s}) := by
have :
(Function.uncurry fun a b => η (a, b) {c : γ | (b, c) ∈ s}) = fun p =>
η p {c : γ | (p.2, c) ∈ s} := by
ext1 p
rw [Function.uncurry_apply_pair]
rw [this]
exact measurable_kernel_prod_mk_left (measurable_fst.snd.prod_mk measurable_snd hs) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η hs : MeasurableSet s h_meas : Measurable (Function.uncurry fun a b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s}) ⊢ Measurable fun a => ∫⁻ (b : β), ↑↑(↑η (a, b)) {c | (b, c) ∈ s} ∂↑κ a ** exact h_meas.lintegral_kernel_prod_right ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η hs : MeasurableSet s ⊢ Measurable (Function.uncurry fun a b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s}) ** have :
(Function.uncurry fun a b => η (a, b) {c : γ | (b, c) ∈ s}) = fun p =>
η p {c : γ | (p.2, c) ∈ s} := by
ext1 p
rw [Function.uncurry_apply_pair] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η hs : MeasurableSet s this : (Function.uncurry fun a b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s}) = fun p => ↑↑(↑η p) {c | (p.2, c) ∈ s} ⊢ Measurable (Function.uncurry fun a b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s}) ** rw [this] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η hs : MeasurableSet s this : (Function.uncurry fun a b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s}) = fun p => ↑↑(↑η p) {c | (p.2, c) ∈ s} ⊢ Measurable fun p => ↑↑(↑η p) {c | (p.2, c) ∈ s} ** exact measurable_kernel_prod_mk_left (measurable_fst.snd.prod_mk measurable_snd hs) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η hs : MeasurableSet s ⊢ (Function.uncurry fun a b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s}) = fun p => ↑↑(↑η p) {c | (p.2, c) ∈ s} ** ext1 p ** case h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η hs : MeasurableSet s p : α × β ⊢ Function.uncurry (fun a b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s}) p = ↑↑(↑η p) {c | (p.2, c) ∈ s} ** rw [Function.uncurry_apply_pair] ** Qed
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ProbabilityTheory.kernel.measurable_compProdFun ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s ⊢ Measurable fun a => compProdFun κ η a s ** simp_rw [compProdFun_tsum_right κ η _ hs] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s ⊢ Measurable fun a => ∑' (n : ℕ), compProdFun κ (seq η n) a s ** refine' Measurable.ennreal_tsum fun n => _ ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s n : ℕ ⊢ Measurable fun a => compProdFun κ (seq η n) a s ** simp only [compProdFun] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s n : ℕ ⊢ Measurable fun a => ∫⁻ (b : β), ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a ** have h_meas : Measurable (Function.uncurry fun a b => seq η n (a, b) {c : γ | (b, c) ∈ s}) := by
have :
(Function.uncurry fun a b => seq η n (a, b) {c : γ | (b, c) ∈ s}) = fun p =>
seq η n p {c : γ | (p.2, c) ∈ s} := by
ext1 p
rw [Function.uncurry_apply_pair]
rw [this]
exact measurable_kernel_prod_mk_left (measurable_fst.snd.prod_mk measurable_snd hs) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s n : ℕ h_meas : Measurable (Function.uncurry fun a b => ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s}) ⊢ Measurable fun a => ∫⁻ (b : β), ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s} ∂↑κ a ** exact h_meas.lintegral_kernel_prod_right ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s n : ℕ ⊢ Measurable (Function.uncurry fun a b => ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s}) ** have :
(Function.uncurry fun a b => seq η n (a, b) {c : γ | (b, c) ∈ s}) = fun p =>
seq η n p {c : γ | (p.2, c) ∈ s} := by
ext1 p
rw [Function.uncurry_apply_pair] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s n : ℕ this : (Function.uncurry fun a b => ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s}) = fun p => ↑↑(↑(seq η n) p) {c | (p.2, c) ∈ s} ⊢ Measurable (Function.uncurry fun a b => ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s}) ** rw [this] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s n : ℕ this : (Function.uncurry fun a b => ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s}) = fun p => ↑↑(↑(seq η n) p) {c | (p.2, c) ∈ s} ⊢ Measurable fun p => ↑↑(↑(seq η n) p) {c | (p.2, c) ∈ s} ** exact measurable_kernel_prod_mk_left (measurable_fst.snd.prod_mk measurable_snd hs) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s n : ℕ ⊢ (Function.uncurry fun a b => ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s}) = fun p => ↑↑(↑(seq η n) p) {c | (p.2, c) ∈ s} ** ext1 p ** case h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η hs : MeasurableSet s n : ℕ p : α × β ⊢ Function.uncurry (fun a b => ↑↑(↑(seq η n) (a, b)) {c | (b, c) ∈ s}) p = ↑↑(↑(seq η n) p) {c | (p.2, c) ∈ s} ** rw [Function.uncurry_apply_pair] ** Qed
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ProbabilityTheory.kernel.compProd_of_not_isSFiniteKernel_left ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } h : ¬IsSFiniteKernel κ ⊢ κ ⊗ₖ η = 0 ** rw [compProd, dif_neg] ** case hnc α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } h : ¬IsSFiniteKernel κ ⊢ ¬(IsSFiniteKernel κ ∧ IsSFiniteKernel η) ** simp [h] ** Qed
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ProbabilityTheory.kernel.compProd_of_not_isSFiniteKernel_right ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } h : ¬IsSFiniteKernel η ⊢ κ ⊗ₖ η = 0 ** rw [compProd, dif_neg] ** case hnc α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } h : ¬IsSFiniteKernel η ⊢ ¬(IsSFiniteKernel κ ∧ IsSFiniteKernel η) ** simp [h] ** Qed
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ProbabilityTheory.kernel.ae_kernel_lt_top ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ ⊢ ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) < ⊤ ** let t := toMeasurable ((κ ⊗ₖ η) a) s ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ t : Set (β × γ) := toMeasurable (↑(κ ⊗ₖ η) a) s ⊢ ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) < ⊤ ** have : ∀ b : β, η (a, b) (Prod.mk b ⁻¹' s) ≤ η (a, b) (Prod.mk b ⁻¹' t) := fun b =>
measure_mono (Set.preimage_mono (subset_toMeasurable _ _)) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ t : Set (β × γ) := toMeasurable (↑(κ ⊗ₖ η) a) s this : ∀ (b : β), ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) ≤ ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) ⊢ ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) < ⊤ ** have ht : MeasurableSet t := measurableSet_toMeasurable _ _ ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ t : Set (β × γ) := toMeasurable (↑(κ ⊗ₖ η) a) s this : ∀ (b : β), ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) ≤ ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) ht : MeasurableSet t ⊢ ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) < ⊤ ** have h2t : (κ ⊗ₖ η) a t ≠ ∞ := by rwa [measure_toMeasurable] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ t : Set (β × γ) := toMeasurable (↑(κ ⊗ₖ η) a) s this : ∀ (b : β), ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) ≤ ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) ht : MeasurableSet t h2t : ↑↑(↑(κ ⊗ₖ η) a) t ≠ ⊤ ⊢ ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) < ⊤ ** have ht_lt_top : ∀ᵐ b ∂κ a, η (a, b) (Prod.mk b ⁻¹' t) < ∞ := by
rw [kernel.compProd_apply _ _ _ ht] at h2t
exact ae_lt_top (kernel.measurable_kernel_prod_mk_left' ht a) h2t ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ t : Set (β × γ) := toMeasurable (↑(κ ⊗ₖ η) a) s this : ∀ (b : β), ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) ≤ ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) ht : MeasurableSet t h2t : ↑↑(↑(κ ⊗ₖ η) a) t ≠ ⊤ ht_lt_top : ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) < ⊤ ⊢ ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) < ⊤ ** filter_upwards [ht_lt_top] with b hb ** case h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ t : Set (β × γ) := toMeasurable (↑(κ ⊗ₖ η) a) s this : ∀ (b : β), ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) ≤ ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) ht : MeasurableSet t h2t : ↑↑(↑(κ ⊗ₖ η) a) t ≠ ⊤ ht_lt_top : ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) < ⊤ b : β hb : ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) < ⊤ ⊢ ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) < ⊤ ** exact (this b).trans_lt hb ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ t : Set (β × γ) := toMeasurable (↑(κ ⊗ₖ η) a) s this : ∀ (b : β), ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) ≤ ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) ht : MeasurableSet t ⊢ ↑↑(↑(κ ⊗ₖ η) a) t ≠ ⊤ ** rwa [measure_toMeasurable] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ t : Set (β × γ) := toMeasurable (↑(κ ⊗ₖ η) a) s this : ∀ (b : β), ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) ≤ ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) ht : MeasurableSet t h2t : ↑↑(↑(κ ⊗ₖ η) a) t ≠ ⊤ ⊢ ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) < ⊤ ** rw [kernel.compProd_apply _ _ _ ht] at h2t ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α h2s : ↑↑(↑(κ ⊗ₖ η) a) s ≠ ⊤ t : Set (β × γ) := toMeasurable (↑(κ ⊗ₖ η) a) s this : ∀ (b : β), ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) ≤ ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) ht : MeasurableSet t h2t : ∫⁻ (b : β), ↑↑(↑η (a, b)) {c | (b, c) ∈ t} ∂↑κ a ≠ ⊤ ⊢ ∀ᵐ (b : β) ∂↑κ a, ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' t) < ⊤ ** exact ae_lt_top (kernel.measurable_kernel_prod_mk_left' ht a) h2t ** Qed
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ProbabilityTheory.kernel.compProd_null ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α hs : MeasurableSet s ⊢ ↑↑(↑(κ ⊗ₖ η) a) s = 0 ↔ (fun b => ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s)) =ᵐ[↑κ a] 0 ** rw [kernel.compProd_apply _ _ _ hs, lintegral_eq_zero_iff] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α hs : MeasurableSet s ⊢ (fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s}) =ᵐ[↑κ a] 0 ↔ (fun b => ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s)) =ᵐ[↑κ a] 0 ** rfl ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ a : α hs : MeasurableSet s ⊢ Measurable fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s} ** exact kernel.measurable_kernel_prod_mk_left' hs a ** Qed
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ProbabilityTheory.kernel.compProd_restrict_left ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α s : Set β hs : MeasurableSet s ⊢ kernel.restrict κ hs ⊗ₖ η = kernel.restrict (κ ⊗ₖ η) (_ : MeasurableSet (s ×ˢ Set.univ)) ** rw [← compProd_restrict] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α s : Set β hs : MeasurableSet s ⊢ kernel.restrict κ hs ⊗ₖ η = kernel.restrict κ ?hs ⊗ₖ kernel.restrict η ?ht case hs α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α s : Set β hs : MeasurableSet s ⊢ MeasurableSet s case ht α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α s : Set β hs : MeasurableSet s ⊢ MeasurableSet Set.univ ** congr ** case e_η α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α s : Set β hs : MeasurableSet s ⊢ η = kernel.restrict η ?ht case ht α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α s : Set β hs : MeasurableSet s ⊢ MeasurableSet Set.univ ** exact kernel.restrict_univ.symm ** Qed
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ProbabilityTheory.kernel.compProd_restrict_right ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α t : Set γ ht : MeasurableSet t ⊢ κ ⊗ₖ kernel.restrict η ht = kernel.restrict (κ ⊗ₖ η) (_ : MeasurableSet (Set.univ ×ˢ t)) ** rw [← compProd_restrict] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α t : Set γ ht : MeasurableSet t ⊢ κ ⊗ₖ kernel.restrict η ht = kernel.restrict κ ?hs ⊗ₖ kernel.restrict η ?ht case hs α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α t : Set γ ht : MeasurableSet t ⊢ MeasurableSet Set.univ case ht α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α t : Set γ ht : MeasurableSet t ⊢ MeasurableSet t ** congr ** case e_κ α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α t : Set γ ht : MeasurableSet t ⊢ κ = kernel.restrict κ ?hs case hs α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α t : Set γ ht : MeasurableSet t ⊢ MeasurableSet Set.univ ** exact kernel.restrict_univ.symm ** Qed
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ProbabilityTheory.kernel.lintegral_compProd' ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) ⊢ ∫⁻ (bc : β × γ), f bc.1 bc.2 ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f b c ∂↑η (a, b) ∂↑κ a ** let F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) ⊢ ∫⁻ (bc : β × γ), f bc.1 bc.2 ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f b c ∂↑η (a, b) ∂↑κ a ** have h : ∀ a, ⨆ n, F n a = Function.uncurry f a :=
SimpleFunc.iSup_eapprox_apply (Function.uncurry f) hf ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β × γ), ⨆ n, ↑(F n) a = Function.uncurry f a ⊢ ∫⁻ (bc : β × γ), f bc.1 bc.2 ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f b c ∂↑η (a, b) ∂↑κ a ** simp only [Prod.forall, Function.uncurry_apply_pair] at h ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b ⊢ ∫⁻ (bc : β × γ), f bc.1 bc.2 ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f b c ∂↑η (a, b) ∂↑κ a ** simp_rw [← h] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b ⊢ ∫⁻ (bc : β × γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (bc.1, bc.2) ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (b, c) ∂↑η (a, b) ∂↑κ a ** have h_mono : Monotone F := fun i j hij b =>
SimpleFunc.monotone_eapprox (Function.uncurry f) hij _ ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F ⊢ ∫⁻ (bc : β × γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (bc.1, bc.2) ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (b, c) ∂↑η (a, b) ∂↑κ a ** rw [lintegral_iSup (fun n => (F n).measurable) h_mono] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) ⊢ ⨆ n, ∫⁻ (a : β × γ), ↑(F n) a ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (b, c) ∂↑η (a, b) ∂↑κ a ** simp_rw [this] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) ⊢ ⨆ n, ∫⁻ (a : β × γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) a ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ⨆ n, ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (b, c) ∂↑η (a, b) ∂↑κ a ** have h_some_meas_integral :
∀ f' : SimpleFunc (β × γ) ℝ≥0∞, Measurable fun b => ∫⁻ c, f' (b, c) ∂η (a, b) := by
intro f'
have :
(fun b => ∫⁻ c, f' (b, c) ∂η (a, b)) =
(fun ab => ∫⁻ c, f' (ab.2, c) ∂η ab) ∘ fun b => (a, b) :=
by ext1 ab; rfl
rw [this]
refine' Measurable.comp _ measurable_prod_mk_left
exact
Measurable.lintegral_kernel_prod_right
((SimpleFunc.measurable _).comp (measurable_fst.snd.prod_mk measurable_snd)) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ⊢ ⨆ n, ∫⁻ (a : β × γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) a ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ⨆ n, ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (b, c) ∂↑η (a, b) ∂↑κ a ** rw [lintegral_iSup] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ⊢ ⨆ n, ∫⁻ (a : β × γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) a ∂↑(κ ⊗ₖ η) a = ⨆ n, ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a_1, c) ∂↑η (a, a_1) ∂↑κ a case hf α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ⊢ ∀ (n : ℕ), Measurable fun b => ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (b, c) ∂↑η (a, b) case h_mono α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ⊢ Monotone fun n b => ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (b, c) ∂↑η (a, b) ** rotate_left ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ⊢ ⨆ n, ∫⁻ (a : β × γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) a ∂↑(κ ⊗ₖ η) a = ⨆ n, ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a_1, c) ∂↑η (a, a_1) ∂↑κ a ** congr ** case e_s α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ⊢ (fun n => ∫⁻ (a : β × γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) a ∂↑(κ ⊗ₖ η) a) = fun n => ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a_1, c) ∂↑η (a, a_1) ∂↑κ a ** ext1 n ** case e_s.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ ⊢ ∫⁻ (a : β × γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a_1, c) ∂↑η (a, a_1) ∂↑κ a ** refine' SimpleFunc.induction (P := fun f => (∫⁻ (a : β × γ), f a ∂(κ ⊗ₖ η) a =
∫⁻ (a_1 : β), ∫⁻ (c : γ), f (a_1, c) ∂η (a, a_1) ∂κ a)) _ _ (F n) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F ⊢ ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) ** intro a ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F a : β ⊢ ∫⁻ (c : γ), ⨆ n, ↑(F n) (a, c) ∂↑η (a✝, a) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (a, c) ∂↑η (a✝, a) ** rw [lintegral_iSup] ** case hf α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F a : β ⊢ ∀ (n : ℕ), Measurable fun c => ↑(F n) (a, c) ** exact fun n => (F n).measurable.comp measurable_prod_mk_left ** case h_mono α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a✝ : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F a : β ⊢ Monotone fun n c => ↑(F n) (a, c) ** exact fun i j hij b => h_mono hij _ ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) ⊢ ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ** intro f' ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) f' : SimpleFunc (β × γ) ℝ≥0∞ ⊢ Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ** have :
(fun b => ∫⁻ c, f' (b, c) ∂η (a, b)) =
(fun ab => ∫⁻ c, f' (ab.2, c) ∂η ab) ∘ fun b => (a, b) :=
by ext1 ab; rfl ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this✝ : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) f' : SimpleFunc (β × γ) ℝ≥0∞ this : (fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b)) = (fun ab => ∫⁻ (c : γ), ↑f' (ab.2, c) ∂↑η ab) ∘ fun b => (a, b) ⊢ Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ** rw [this] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this✝ : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) f' : SimpleFunc (β × γ) ℝ≥0∞ this : (fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b)) = (fun ab => ∫⁻ (c : γ), ↑f' (ab.2, c) ∂↑η ab) ∘ fun b => (a, b) ⊢ Measurable ((fun ab => ∫⁻ (c : γ), ↑f' (ab.2, c) ∂↑η ab) ∘ fun b => (a, b)) ** refine' Measurable.comp _ measurable_prod_mk_left ** case refine'_1 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this✝ : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) f' : SimpleFunc (β × γ) ℝ≥0∞ this : (fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b)) = (fun ab => ∫⁻ (c : γ), ↑f' (ab.2, c) ∂↑η ab) ∘ fun b => (a, b) ⊢ Measurable fun ab => ∫⁻ (c : γ), ↑f' (ab.2, c) ∂↑η ab case refine'_2 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this✝ : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) f' : SimpleFunc (β × γ) ℝ≥0∞ this : (fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b)) = (fun ab => ∫⁻ (c : γ), ↑f' (ab.2, c) ∂↑η ab) ∘ fun b => (a, b) ⊢ MeasurableSpace α ** exact
Measurable.lintegral_kernel_prod_right
((SimpleFunc.measurable _).comp (measurable_fst.snd.prod_mk measurable_snd)) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) f' : SimpleFunc (β × γ) ℝ≥0∞ ⊢ (fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b)) = (fun ab => ∫⁻ (c : γ), ↑f' (ab.2, c) ∂↑η ab) ∘ fun b => (a, b) ** ext1 ab ** case h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) f' : SimpleFunc (β × γ) ℝ≥0∞ ab : β ⊢ ∫⁻ (c : γ), ↑f' (ab, c) ∂↑η (a, ab) = ((fun ab => ∫⁻ (c : γ), ↑f' (ab.2, c) ∂↑η ab) ∘ fun b => (a, b)) ab ** rfl ** case hf α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ⊢ ∀ (n : ℕ), Measurable fun b => ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (b, c) ∂↑η (a, b) ** exact fun n => h_some_meas_integral (F n) ** case h_mono α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) ⊢ Monotone fun n b => ∫⁻ (c : γ), ↑(SimpleFunc.eapprox (Function.uncurry f) n) (b, c) ∂↑η (a, b) ** exact fun i j hij b => lintegral_mono fun c => h_mono hij _ ** case e_s.h.refine'_1 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ ⊢ ∀ (c : ℝ≥0∞) {s : Set (β × γ)} (hs : MeasurableSet s), (fun f => ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a) (SimpleFunc.piecewise s hs (SimpleFunc.const (β × γ) c) (SimpleFunc.const (β × γ) 0)) ** intro c s hs ** case e_s.h.refine'_1 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ c : ℝ≥0∞ s : Set (β × γ) hs : MeasurableSet s ⊢ ∫⁻ (a : β × γ), ↑(SimpleFunc.piecewise s hs (SimpleFunc.const (β × γ) c) (SimpleFunc.const (β × γ) 0)) a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c_1 : γ), ↑(SimpleFunc.piecewise s hs (SimpleFunc.const (β × γ) c) (SimpleFunc.const (β × γ) 0)) (a_1, c_1) ∂↑η (a, a_1) ∂↑κ a ** simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,
SimpleFunc.coe_zero, Set.piecewise_eq_indicator, Function.const, lintegral_indicator_const hs] ** case e_s.h.refine'_1 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ c : ℝ≥0∞ s : Set (β × γ) hs : MeasurableSet s ⊢ c * ↑↑(↑(κ ⊗ₖ η) a) s = ∫⁻ (a_1 : β), ∫⁻ (c_1 : γ), Set.indicator s (fun x => c) (a_1, c_1) ∂↑η (a, a_1) ∂↑κ a ** rw [compProd_apply κ η _ hs, ← lintegral_const_mul c _] ** case e_s.h.refine'_1 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ c : ℝ≥0∞ s : Set (β × γ) hs : MeasurableSet s ⊢ ∫⁻ (a_1 : β), c * ↑↑(↑η (a, a_1)) {c | (a_1, c) ∈ s} ∂↑κ a = ∫⁻ (a_1 : β), ∫⁻ (c_1 : γ), Set.indicator s (fun x => c) (a_1, c_1) ∂↑η (a, a_1) ∂↑κ a α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ c : ℝ≥0∞ s : Set (β × γ) hs : MeasurableSet s ⊢ Measurable fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s} ** swap ** case e_s.h.refine'_1 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ c : ℝ≥0∞ s : Set (β × γ) hs : MeasurableSet s ⊢ ∫⁻ (a_1 : β), c * ↑↑(↑η (a, a_1)) {c | (a_1, c) ∈ s} ∂↑κ a = ∫⁻ (a_1 : β), ∫⁻ (c_1 : γ), Set.indicator s (fun x => c) (a_1, c_1) ∂↑η (a, a_1) ∂↑κ a ** congr ** case e_s.h.refine'_1.e_f α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ c : ℝ≥0∞ s : Set (β × γ) hs : MeasurableSet s ⊢ (fun a_1 => c * ↑↑(↑η (a, a_1)) {c | (a_1, c) ∈ s}) = fun a_1 => ∫⁻ (c_1 : γ), Set.indicator s (fun x => c) (a_1, c_1) ∂↑η (a, a_1) ** ext1 b ** case e_s.h.refine'_1.e_f.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ c : ℝ≥0∞ s : Set (β × γ) hs : MeasurableSet s b : β ⊢ c * ↑↑(↑η (a, b)) {c | (b, c) ∈ s} = ∫⁻ (c_1 : γ), Set.indicator s (fun x => c) (b, c_1) ∂↑η (a, b) ** rw [lintegral_indicator_const_comp measurable_prod_mk_left hs] ** case e_s.h.refine'_1.e_f.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ c : ℝ≥0∞ s : Set (β × γ) hs : MeasurableSet s b : β ⊢ c * ↑↑(↑η (a, b)) {c | (b, c) ∈ s} = c * ↑↑(↑η (a, b)) (Prod.mk b ⁻¹' s) ** rfl ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ c : ℝ≥0∞ s : Set (β × γ) hs : MeasurableSet s ⊢ Measurable fun b => ↑↑(↑η (a, b)) {c | (b, c) ∈ s} ** exact (measurable_kernel_prod_mk_left ((measurable_fst.snd.prod_mk measurable_snd) hs)).comp
measurable_prod_mk_left ** case e_s.h.refine'_2 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f) n) (a, b) = f a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ ⊢ ∀ ⦃f g : SimpleFunc (β × γ) ℝ≥0∞⦄, Disjoint (Function.support ↑f) (Function.support ↑g) → (fun f => ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a) f → (fun f => ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a) g → (fun f => ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a) (f + g) ** intro f f' _ hf_eq hf'_eq ** case e_s.h.refine'_2 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f✝ : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f✝) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ f f' : SimpleFunc (β × γ) ℝ≥0∞ a✝ : Disjoint (Function.support ↑f) (Function.support ↑f') hf_eq : ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a ⊢ ∫⁻ (a : β × γ), ↑(f + f') a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑(f + f') (a_1, c) ∂↑η (a, a_1) ∂↑κ a ** simp_rw [SimpleFunc.coe_add, Pi.add_apply] ** case e_s.h.refine'_2 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f✝ : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f✝) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ f f' : SimpleFunc (β × γ) ℝ≥0∞ a✝ : Disjoint (Function.support ↑f) (Function.support ↑f') hf_eq : ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a ⊢ ∫⁻ (a : β × γ), ↑f a + ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) + ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a ** change
∫⁻ x, (f : β × γ → ℝ≥0∞) x + f' x ∂(κ ⊗ₖ η) a =
∫⁻ b, ∫⁻ c : γ, f (b, c) + f' (b, c) ∂η (a, b) ∂κ a ** case e_s.h.refine'_2 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f✝ : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f✝) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ f f' : SimpleFunc (β × γ) ℝ≥0∞ a✝ : Disjoint (Function.support ↑f) (Function.support ↑f') hf_eq : ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a ⊢ ∫⁻ (x : β × γ), ↑f x + ↑f' x ∂↑(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ↑f (b, c) + ↑f' (b, c) ∂↑η (a, b) ∂↑κ a ** rw [lintegral_add_left (SimpleFunc.measurable _), hf_eq, hf'_eq, ← lintegral_add_left] ** case e_s.h.refine'_2 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f✝ : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f✝) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ f f' : SimpleFunc (β × γ) ℝ≥0∞ a✝ : Disjoint (Function.support ↑f) (Function.support ↑f') hf_eq : ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a ⊢ ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) + ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a = ∫⁻ (b : β), ∫⁻ (c : γ), ↑f (b, c) + ↑f' (b, c) ∂↑η (a, b) ∂↑κ a case e_s.h.refine'_2.hf α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f✝ : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f✝) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ f f' : SimpleFunc (β × γ) ℝ≥0∞ a✝ : Disjoint (Function.support ↑f) (Function.support ↑f') hf_eq : ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a ⊢ Measurable fun a_1 => ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ** swap ** case e_s.h.refine'_2 α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f✝ : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f✝) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ f f' : SimpleFunc (β × γ) ℝ≥0∞ a✝ : Disjoint (Function.support ↑f) (Function.support ↑f') hf_eq : ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a ⊢ ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) + ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a = ∫⁻ (b : β), ∫⁻ (c : γ), ↑f (b, c) + ↑f' (b, c) ∂↑η (a, b) ∂↑κ a ** congr with b ** case e_s.h.refine'_2.e_f.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f✝ : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f✝) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ f f' : SimpleFunc (β × γ) ℝ≥0∞ a✝ : Disjoint (Function.support ↑f) (Function.support ↑f') hf_eq : ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a b : β ⊢ ∫⁻ (c : γ), ↑f (b, c) ∂↑η (a, b) + ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) = ∫⁻ (c : γ), ↑f (b, c) + ↑f' (b, c) ∂↑η (a, b) ** rw [lintegral_add_left] ** case e_s.h.refine'_2.e_f.h.hf α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f✝ : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f✝) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ f f' : SimpleFunc (β × γ) ℝ≥0∞ a✝ : Disjoint (Function.support ↑f) (Function.support ↑f') hf_eq : ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a b : β ⊢ Measurable fun c => ↑f (b, c) ** exact (SimpleFunc.measurable _).comp measurable_prod_mk_left ** case e_s.h.refine'_2.hf α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f✝ : β → γ → ℝ≥0∞ hf : Measurable (Function.uncurry f✝) F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝) h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b h_mono : Monotone F this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂↑η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂↑η (a, b) h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂↑η (a, b) n : ℕ f f' : SimpleFunc (β × γ) ℝ≥0∞ a✝ : Disjoint (Function.support ↑f) (Function.support ↑f') hf_eq : ∫⁻ (a : β × γ), ↑f a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ∂↑κ a hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂↑(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂↑η (a, a_1) ∂↑κ a ⊢ Measurable fun a_1 => ∫⁻ (c : γ), ↑f (a_1, c) ∂↑η (a, a_1) ** exact h_some_meas_integral f ** Qed
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ProbabilityTheory.kernel.lintegral_compProd₀ ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → ℝ≥0∞ hf : AEMeasurable f ⊢ ∫⁻ (z : β × γ), f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (x : β), ∫⁻ (y : γ), f (x, y) ∂↑η (a, x) ∂↑κ a ** have A : ∫⁻ z, f z ∂(κ ⊗ₖ η) a = ∫⁻ z, hf.mk f z ∂(κ ⊗ₖ η) a := lintegral_congr_ae hf.ae_eq_mk ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (z : β × γ), f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (z : β × γ), AEMeasurable.mk f hf z ∂↑(κ ⊗ₖ η) a ⊢ ∫⁻ (z : β × γ), f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (x : β), ∫⁻ (y : γ), f (x, y) ∂↑η (a, x) ∂↑κ a ** have B : ∫⁻ x, ∫⁻ y, f (x, y) ∂η (a, x) ∂κ a = ∫⁻ x, ∫⁻ y, hf.mk f (x, y) ∂η (a, x) ∂κ a := by
apply lintegral_congr_ae
filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with _ ha using lintegral_congr_ae ha ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (z : β × γ), f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (z : β × γ), AEMeasurable.mk f hf z ∂↑(κ ⊗ₖ η) a B : ∫⁻ (x : β), ∫⁻ (y : γ), f (x, y) ∂↑η (a, x) ∂↑κ a = ∫⁻ (x : β), ∫⁻ (y : γ), AEMeasurable.mk f hf (x, y) ∂↑η (a, x) ∂↑κ a ⊢ ∫⁻ (z : β × γ), f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (x : β), ∫⁻ (y : γ), f (x, y) ∂↑η (a, x) ∂↑κ a ** rw [A, B, lintegral_compProd] ** case hf α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (z : β × γ), f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (z : β × γ), AEMeasurable.mk f hf z ∂↑(κ ⊗ₖ η) a B : ∫⁻ (x : β), ∫⁻ (y : γ), f (x, y) ∂↑η (a, x) ∂↑κ a = ∫⁻ (x : β), ∫⁻ (y : γ), AEMeasurable.mk f hf (x, y) ∂↑η (a, x) ∂↑κ a ⊢ Measurable fun z => AEMeasurable.mk f hf z ** exact hf.measurable_mk ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (z : β × γ), f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (z : β × γ), AEMeasurable.mk f hf z ∂↑(κ ⊗ₖ η) a ⊢ ∫⁻ (x : β), ∫⁻ (y : γ), f (x, y) ∂↑η (a, x) ∂↑κ a = ∫⁻ (x : β), ∫⁻ (y : γ), AEMeasurable.mk f hf (x, y) ∂↑η (a, x) ∂↑κ a ** apply lintegral_congr_ae ** case h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (z : β × γ), f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (z : β × γ), AEMeasurable.mk f hf z ∂↑(κ ⊗ₖ η) a ⊢ (fun a_1 => ∫⁻ (y : γ), f (a_1, y) ∂↑η (a, a_1)) =ᵐ[↑κ a] fun a_1 => ∫⁻ (y : γ), AEMeasurable.mk f hf (a_1, y) ∂↑η (a, a_1) ** filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with _ ha using lintegral_congr_ae ha ** Qed
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ProbabilityTheory.kernel.set_lintegral_compProd ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → ℝ≥0∞ hf : Measurable f s : Set β t : Set γ hs : MeasurableSet s ht : MeasurableSet t ⊢ ∫⁻ (z : β × γ) in s ×ˢ t, f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (x : β) in s, ∫⁻ (y : γ) in t, f (x, y) ∂↑η (a, x) ∂↑κ a ** simp_rw [← kernel.restrict_apply (κ ⊗ₖ η) (hs.prod ht), ← compProd_restrict hs ht,
lintegral_compProd _ _ _ hf, kernel.restrict_apply] ** Qed
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ProbabilityTheory.kernel.set_lintegral_compProd_univ_right ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s✝ : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → ℝ≥0∞ hf : Measurable f s : Set β hs : MeasurableSet s ⊢ ∫⁻ (z : β × γ) in s ×ˢ Set.univ, f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (x : β) in s, ∫⁻ (y : γ), f (x, y) ∂↑η (a, x) ∂↑κ a ** simp_rw [set_lintegral_compProd κ η a hf hs MeasurableSet.univ, Measure.restrict_univ] ** Qed
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ProbabilityTheory.kernel.set_lintegral_compProd_univ_left ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → ℝ≥0∞ hf : Measurable f t : Set γ ht : MeasurableSet t ⊢ ∫⁻ (z : β × γ) in Set.univ ×ˢ t, f z ∂↑(κ ⊗ₖ η) a = ∫⁻ (x : β), ∫⁻ (y : γ) in t, f (x, y) ∂↑η (a, x) ∂↑κ a ** simp_rw [set_lintegral_compProd κ η a hf MeasurableSet.univ ht, Measure.restrict_univ] ** Qed
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ProbabilityTheory.kernel.compProd_apply_univ_le ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η a : α ⊢ ↑↑(↑(κ ⊗ₖ η) a) Set.univ ≤ ↑↑(↑κ a) Set.univ * IsFiniteKernel.bound η ** by_cases hκ : IsSFiniteKernel κ ** case pos α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η a : α hκ : IsSFiniteKernel κ ⊢ ↑↑(↑(κ ⊗ₖ η) a) Set.univ ≤ ↑↑(↑κ a) Set.univ * IsFiniteKernel.bound η case neg α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η a : α hκ : ¬IsSFiniteKernel κ ⊢ ↑↑(↑(κ ⊗ₖ η) a) Set.univ ≤ ↑↑(↑κ a) Set.univ * IsFiniteKernel.bound η ** swap ** case pos α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η a : α hκ : IsSFiniteKernel κ ⊢ ↑↑(↑(κ ⊗ₖ η) a) Set.univ ≤ ↑↑(↑κ a) Set.univ * IsFiniteKernel.bound η ** rw [compProd_apply κ η a MeasurableSet.univ] ** case pos α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η a : α hκ : IsSFiniteKernel κ ⊢ ∫⁻ (b : β), ↑↑(↑η (a, b)) {c | (b, c) ∈ Set.univ} ∂↑κ a ≤ ↑↑(↑κ a) Set.univ * IsFiniteKernel.bound η ** simp only [Set.mem_univ, Set.setOf_true] ** case pos α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η a : α hκ : IsSFiniteKernel κ ⊢ ∫⁻ (b : β), ↑↑(↑η (a, b)) Set.univ ∂↑κ a ≤ ↑↑(↑κ a) Set.univ * IsFiniteKernel.bound η ** let Cη := IsFiniteKernel.bound η ** case pos α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η a : α hκ : IsSFiniteKernel κ Cη : ℝ≥0∞ := IsFiniteKernel.bound η ⊢ ∫⁻ (b : β), ↑↑(↑η (a, b)) Set.univ ∂↑κ a ≤ ↑↑(↑κ a) Set.univ * IsFiniteKernel.bound η ** calc
∫⁻ b, η (a, b) Set.univ ∂κ a ≤ ∫⁻ _, Cη ∂κ a :=
lintegral_mono fun b => measure_le_bound η (a, b) Set.univ
_ = Cη * κ a Set.univ := (MeasureTheory.lintegral_const Cη)
_ = κ a Set.univ * Cη := mul_comm _ _ ** case neg α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η a : α hκ : ¬IsSFiniteKernel κ ⊢ ↑↑(↑(κ ⊗ₖ η) a) Set.univ ≤ ↑↑(↑κ a) Set.univ * IsFiniteKernel.bound η ** rw [compProd_of_not_isSFiniteKernel_left _ _ hκ] ** case neg α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ s : Set (β × γ) κ : { x // x ∈ kernel α β } η : { x // x ∈ kernel (α × β) γ } inst✝ : IsFiniteKernel η a : α hκ : ¬IsSFiniteKernel κ ⊢ ↑↑(↑0 a) Set.univ ≤ ↑↑(↑κ a) Set.univ * IsFiniteKernel.bound η ** simp ** Qed
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ProbabilityTheory.kernel.map_apply' ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } hf : Measurable f a : α s : Set γ hs : MeasurableSet s ⊢ ↑↑(↑(map κ f hf) a) s = ↑↑(↑κ a) (f ⁻¹' s) ** rw [map_apply, Measure.map_apply hf hs] ** Qed
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ProbabilityTheory.kernel.lintegral_map ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } hf : Measurable f a : α g' : γ → ℝ≥0∞ hg : Measurable g' ⊢ ∫⁻ (b : γ), g' b ∂↑(map κ f hf) a = ∫⁻ (a : β), g' (f a) ∂↑κ a ** rw [map_apply _ hf, lintegral_map hg hf] ** Qed
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ProbabilityTheory.kernel.sum_map_seq ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } inst✝ : IsSFiniteKernel κ hf : Measurable f ⊢ (kernel.sum fun n => map (seq κ n) f hf) = map κ f hf ** ext a s hs ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } inst✝ : IsSFiniteKernel κ hf : Measurable f a : α s : Set γ hs : MeasurableSet s ⊢ ↑↑(↑(kernel.sum fun n => map (seq κ n) f hf) a) s = ↑↑(↑(map κ f hf) a) s ** rw [kernel.sum_apply, map_apply' κ hf a hs, Measure.sum_apply _ hs, ← measure_sum_seq κ,
Measure.sum_apply _ (hf hs)] ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } inst✝ : IsSFiniteKernel κ hf : Measurable f a : α s : Set γ hs : MeasurableSet s ⊢ ∑' (i : ℕ), ↑↑(↑(map (seq κ i) f hf) a) s = ∑' (i : ℕ), ↑↑(↑(seq κ i) a) (f ⁻¹' s) ** simp_rw [map_apply' _ hf _ hs] ** Qed
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ProbabilityTheory.kernel.sum_comap_seq ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } inst✝ : IsSFiniteKernel κ hg : Measurable g ⊢ (kernel.sum fun n => comap (seq κ n) g hg) = comap κ g hg ** ext a s hs ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } inst✝ : IsSFiniteKernel κ hg : Measurable g a : γ s : Set β hs : MeasurableSet s ⊢ ↑↑(↑(kernel.sum fun n => comap (seq κ n) g hg) a) s = ↑↑(↑(comap κ g hg) a) s ** rw [kernel.sum_apply, comap_apply' κ hg a s, Measure.sum_apply _ hs, ← measure_sum_seq κ,
Measure.sum_apply _ hs] ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } inst✝ : IsSFiniteKernel κ hg : Measurable g a : γ s : Set β hs : MeasurableSet s ⊢ ∑' (i : ℕ), ↑↑(↑(comap (seq κ i) g hg) a) s = ∑' (i : ℕ), ↑↑(↑(seq κ i) (g a)) s ** simp_rw [comap_apply' _ hg _ s] ** Qed
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ProbabilityTheory.kernel.lintegral_swapRight ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g✝ : γ → α κ : { x // x ∈ kernel α (β × γ) } a : α g : γ × β → ℝ≥0∞ hg : Measurable g ⊢ ∫⁻ (c : γ × β), g c ∂↑(swapRight κ) a = ∫⁻ (bc : β × γ), g (Prod.swap bc) ∂↑κ a ** rw [swapRight, lintegral_map _ measurable_swap a hg] ** Qed
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ProbabilityTheory.kernel.lintegral_fst ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g✝ : γ → α κ : { x // x ∈ kernel α (β × γ) } a : α g : β → ℝ≥0∞ hg : Measurable g ⊢ ∫⁻ (c : β), g c ∂↑(fst κ) a = ∫⁻ (bc : β × γ), g bc.1 ∂↑κ a ** rw [fst, lintegral_map _ measurable_fst a hg] ** Qed
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ProbabilityTheory.kernel.lintegral_snd ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g✝ : γ → α κ : { x // x ∈ kernel α (β × γ) } a : α g : γ → ℝ≥0∞ hg : Measurable g ⊢ ∫⁻ (c : γ), g c ∂↑(snd κ) a = ∫⁻ (bc : β × γ), g bc.2 ∂↑κ a ** rw [snd, lintegral_map _ measurable_snd a hg] ** Qed
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ProbabilityTheory.kernel.comp_apply' ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α η : { x // x ∈ kernel β γ } κ : { x // x ∈ kernel α β } a : α s : Set γ hs : MeasurableSet s ⊢ ↑↑(↑(η ∘ₖ κ) a) s = ∫⁻ (b : β), ↑↑(↑η b) s ∂↑κ a ** rw [comp_apply, Measure.bind_apply hs (kernel.measurable _)] ** Qed
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ProbabilityTheory.kernel.lintegral_comp ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g✝ : γ → α η : { x // x ∈ kernel β γ } κ : { x // x ∈ kernel α β } a : α g : γ → ℝ≥0∞ hg : Measurable g ⊢ ∫⁻ (c : γ), g c ∂↑(η ∘ₖ κ) a = ∫⁻ (b : β), ∫⁻ (c : γ), g c ∂↑η b ∂↑κ a ** rw [comp_apply, Measure.lintegral_bind (kernel.measurable _) hg] ** Qed
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ProbabilityTheory.kernel.comp_assoc ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α δ : Type u_5 mδ : MeasurableSpace δ ξ : { x // x ∈ kernel γ δ } inst✝ : IsSFiniteKernel ξ η : { x // x ∈ kernel β γ } κ : { x // x ∈ kernel α β } ⊢ ξ ∘ₖ η ∘ₖ κ = ξ ∘ₖ (η ∘ₖ κ) ** refine' ext_fun fun a f hf => _ ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f✝ : β → γ g : γ → α δ : Type u_5 mδ : MeasurableSpace δ ξ : { x // x ∈ kernel γ δ } inst✝ : IsSFiniteKernel ξ η : { x // x ∈ kernel β γ } κ : { x // x ∈ kernel α β } a : α f : δ → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (b : δ), f b ∂↑(ξ ∘ₖ η ∘ₖ κ) a = ∫⁻ (b : δ), f b ∂↑(ξ ∘ₖ (η ∘ₖ κ)) a ** simp_rw [lintegral_comp _ _ _ hf, lintegral_comp _ _ _ hf.lintegral_kernel] ** Qed
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ProbabilityTheory.kernel.deterministic_comp_eq_map ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α hf : Measurable f κ : { x // x ∈ kernel α β } ⊢ deterministic f hf ∘ₖ κ = map κ f hf ** ext a s hs ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α hf : Measurable f κ : { x // x ∈ kernel α β } a : α s : Set γ hs : MeasurableSet s ⊢ ↑↑(↑(deterministic f hf ∘ₖ κ) a) s = ↑↑(↑(map κ f hf) a) s ** simp_rw [map_apply' _ _ _ hs, comp_apply' _ _ _ hs, deterministic_apply' hf _ hs,
lintegral_indicator_const_comp hf hs, one_mul] ** Qed
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ProbabilityTheory.kernel.comp_deterministic_eq_comap ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } hg : Measurable g ⊢ κ ∘ₖ deterministic g hg = comap κ g hg ** ext a s hs ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ f : β → γ g : γ → α κ : { x // x ∈ kernel α β } hg : Measurable g a : γ s : Set β hs : MeasurableSet s ⊢ ↑↑(↑(κ ∘ₖ deterministic g hg) a) s = ↑↑(↑(comap κ g hg) a) s ** simp_rw [comap_apply' _ _ _ s, comp_apply' _ _ _ hs, deterministic_apply hg a,
lintegral_dirac' _ (kernel.measurable_coe κ hs)] ** Qed
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ProbabilityTheory.kernel.prod_apply ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel α γ } inst✝ : IsSFiniteKernel η a : α s : Set (β × γ) hs : MeasurableSet s ⊢ ↑↑(↑(κ ×ₖ η) a) s = ∫⁻ (b : β), ↑↑(↑η a) {c | (b, c) ∈ s} ∂↑κ a ** simp_rw [prod, compProd_apply _ _ _ hs, swapLeft_apply _ _, prodMkLeft_apply, Prod.swap_prod_mk] ** Qed
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ProbabilityTheory.kernel.lintegral_prod ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β γ : Type u_4 mγ : MeasurableSpace γ κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel α γ } inst✝ : IsSFiniteKernel η a : α g : β × γ → ℝ≥0∞ hg : Measurable g ⊢ ∫⁻ (c : β × γ), g c ∂↑(κ ×ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), g (b, c) ∂↑η a ∂↑κ a ** simp_rw [prod, lintegral_compProd _ _ _ hg, swapLeft_apply, prodMkLeft_apply, Prod.swap_prod_mk] ** Qed
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MeasureTheory.pdf_undef ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E X : Ω → E h : ¬HasPDF X ℙ ⊢ pdf X ℙ = 0 ** simp only [pdf, dif_neg h] ** Qed
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MeasureTheory.hasPDF_of_pdf_ne_zero ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E X : Ω → E h : pdf X ℙ ≠ 0 ⊢ HasPDF X ℙ ** by_contra hpdf ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E X : Ω → E h : pdf X ℙ ≠ 0 hpdf : ¬HasPDF X ℙ ⊢ False ** simp [pdf, hpdf] at h ** Qed
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MeasureTheory.measurable_pdf ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : autoParam (Measure E) _auto✝ ⊢ Measurable (pdf X ℙ) ** unfold pdf ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : autoParam (Measure E) _auto✝ ⊢ Measurable (if hX : HasPDF X ℙ then Classical.choose (_ : ∃ f, Measurable f ∧ map X ℙ = withDensity μ f) else 0) ** split_ifs with h ** case pos Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : autoParam (Measure E) _auto✝ h : HasPDF X ℙ ⊢ Measurable (Classical.choose (_ : ∃ f, Measurable f ∧ map X ℙ = withDensity μ f)) case neg Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : autoParam (Measure E) _auto✝ h : ¬HasPDF X ℙ ⊢ Measurable 0 ** exacts [(Classical.choose_spec h.1.2).1, measurable_zero] ** Qed
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MeasureTheory.map_eq_withDensity_pdf ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : autoParam (Measure E) _auto✝ hX : HasPDF X ℙ ⊢ map X ℙ = withDensity μ (pdf X ℙ) ** simp only [pdf, dif_pos hX] ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : autoParam (Measure E) _auto✝ hX : HasPDF X ℙ ⊢ map X ℙ = withDensity μ (Classical.choose (_ : ∃ f, Measurable f ∧ map X ℙ = withDensity μ f)) ** exact (Classical.choose_spec hX.pdf'.2).2 ** Qed
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MeasureTheory.map_eq_set_lintegral_pdf ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : autoParam (Measure E) _auto✝ hX : HasPDF X ℙ s : Set E hs : MeasurableSet s ⊢ ↑↑(map X ℙ) s = ∫⁻ (x : E) in s, pdf X ℙ x ∂μ ** rw [← withDensity_apply _ hs, map_eq_withDensity_pdf X ℙ μ] ** Qed
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MeasureTheory.pdf.lintegral_eq_measure_univ ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E X : Ω → E inst✝ : HasPDF X ℙ ⊢ ∫⁻ (x : E), pdf X ℙ x ∂μ = ↑↑ℙ Set.univ ** rw [← set_lintegral_univ, ← map_eq_set_lintegral_pdf X ℙ μ MeasurableSet.univ,
Measure.map_apply (HasPDF.measurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ] ** Qed
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MeasureTheory.pdf.map_absolutelyContinuous ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E X : Ω → E inst✝ : HasPDF X ℙ ⊢ map X ℙ ≪ μ ** rw [map_eq_withDensity_pdf X ℙ μ] ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E X : Ω → E inst✝ : HasPDF X ℙ ⊢ withDensity μ (pdf X ℙ) ≪ μ ** exact withDensity_absolutelyContinuous _ _ ** Qed
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MeasureTheory.pdf.haveLebesgueDecomposition_of_hasPDF ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E X : Ω → E hX' : HasPDF X ℙ ⊢ Measurable (0, pdf X ℙ).2 ∧ (0, pdf X ℙ).1 ⟂ₘ μ ∧ map X ℙ = (0, pdf X ℙ).1 + withDensity μ (0, pdf X ℙ).2 ** simp only [zero_add, measurable_pdf X ℙ μ, true_and_iff, MutuallySingular.zero_left,
map_eq_withDensity_pdf X ℙ μ] ** Qed
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MeasureTheory.pdf.hasPDF_iff_of_measurable ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E X : Ω → E hX : Measurable X ⊢ HasPDF X ℙ ↔ HaveLebesgueDecomposition (map X ℙ) μ ∧ map X ℙ ≪ μ ** rw [hasPDF_iff] ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E X : Ω → E hX : Measurable X ⊢ Measurable X ∧ HaveLebesgueDecomposition (map X ℙ) μ ∧ map X ℙ ≪ μ ↔ HaveLebesgueDecomposition (map X ℙ) μ ∧ map X ℙ ≪ μ ** simp only [hX, true_and] ** Qed
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MeasureTheory.pdf.quasiMeasurePreserving_hasPDF ** Ω : Type u_1 E : Type u_2 inst✝² : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E F : Type u_3 inst✝¹ : MeasurableSpace F ν : Measure F X : Ω → E inst✝ : HasPDF X ℙ g : E → F hg : QuasiMeasurePreserving g hmap : HaveLebesgueDecomposition (map g (map X ℙ)) ν ⊢ HasPDF (g ∘ X) ℙ ** rw [hasPDF_iff, ← map_map hg.measurable (HasPDF.measurable X ℙ μ)] ** Ω : Type u_1 E : Type u_2 inst✝² : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E F : Type u_3 inst✝¹ : MeasurableSpace F ν : Measure F X : Ω → E inst✝ : HasPDF X ℙ g : E → F hg : QuasiMeasurePreserving g hmap : HaveLebesgueDecomposition (map g (map X ℙ)) ν ⊢ Measurable (g ∘ X) ∧ HaveLebesgueDecomposition (map g (map X ℙ)) ν ∧ map g (map X ℙ) ≪ ν ** refine' ⟨hg.measurable.comp (HasPDF.measurable X ℙ μ), hmap, _⟩ ** Ω : Type u_1 E : Type u_2 inst✝² : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E F : Type u_3 inst✝¹ : MeasurableSpace F ν : Measure F X : Ω → E inst✝ : HasPDF X ℙ g : E → F hg : QuasiMeasurePreserving g hmap : HaveLebesgueDecomposition (map g (map X ℙ)) ν ⊢ map g (map X ℙ) ≪ ν ** rw [map_eq_withDensity_pdf X ℙ μ] ** Ω : Type u_1 E : Type u_2 inst✝² : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E F : Type u_3 inst✝¹ : MeasurableSpace F ν : Measure F X : Ω → E inst✝ : HasPDF X ℙ g : E → F hg : QuasiMeasurePreserving g hmap : HaveLebesgueDecomposition (map g (map X ℙ)) ν ⊢ map g (withDensity μ (pdf X ℙ)) ≪ ν ** refine' AbsolutelyContinuous.mk fun s hsm hs => _ ** Ω : Type u_1 E : Type u_2 inst✝² : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E F : Type u_3 inst✝¹ : MeasurableSpace F ν : Measure F X : Ω → E inst✝ : HasPDF X ℙ g : E → F hg : QuasiMeasurePreserving g hmap : HaveLebesgueDecomposition (map g (map X ℙ)) ν s : Set F hsm : MeasurableSet s hs : ↑↑ν s = 0 ⊢ ↑↑(map g (withDensity μ (pdf X ℙ))) s = 0 ** rw [map_apply hg.measurable hsm, withDensity_apply _ (hg.measurable hsm)] ** Ω : Type u_1 E : Type u_2 inst✝² : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E F : Type u_3 inst✝¹ : MeasurableSpace F ν : Measure F X : Ω → E inst✝ : HasPDF X ℙ g : E → F hg : QuasiMeasurePreserving g hmap : HaveLebesgueDecomposition (map g (map X ℙ)) ν s : Set F hsm : MeasurableSet s hs : ↑↑ν s = 0 ⊢ ∫⁻ (a : E) in g ⁻¹' s, pdf X ℙ a ∂μ = 0 ** have := hg.absolutelyContinuous hs ** Ω : Type u_1 E : Type u_2 inst✝² : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E F : Type u_3 inst✝¹ : MeasurableSpace F ν : Measure F X : Ω → E inst✝ : HasPDF X ℙ g : E → F hg : QuasiMeasurePreserving g hmap : HaveLebesgueDecomposition (map g (map X ℙ)) ν s : Set F hsm : MeasurableSet s hs : ↑↑ν s = 0 this : ↑↑(map g μ) s = 0 ⊢ ∫⁻ (a : E) in g ⁻¹' s, pdf X ℙ a ∂μ = 0 ** rw [map_apply hg.measurable hsm] at this ** Ω : Type u_1 E : Type u_2 inst✝² : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E F : Type u_3 inst✝¹ : MeasurableSpace F ν : Measure F X : Ω → E inst✝ : HasPDF X ℙ g : E → F hg : QuasiMeasurePreserving g hmap : HaveLebesgueDecomposition (map g (map X ℙ)) ν s : Set F hsm : MeasurableSet s hs : ↑↑ν s = 0 this : ↑↑μ (g ⁻¹' s) = 0 ⊢ ∫⁻ (a : E) in g ⁻¹' s, pdf X ℙ a ∂μ = 0 ** exact set_lintegral_measure_zero _ _ this ** Qed
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MeasureTheory.pdf.hasFiniteIntegral_mul ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E inst✝ : IsFiniteMeasure ℙ X : Ω → ℝ f : ℝ → ℝ g : ℝ → ℝ≥0∞ hg : pdf X ℙ =ᶠ[ae volume] g hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤ ⊢ HasFiniteIntegral fun x => f x * ENNReal.toReal (pdf X ℙ x) ** rw [HasFiniteIntegral] ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E inst✝ : IsFiniteMeasure ℙ X : Ω → ℝ f : ℝ → ℝ g : ℝ → ℝ≥0∞ hg : pdf X ℙ =ᶠ[ae volume] g hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤ ⊢ ∫⁻ (a : ℝ), ↑‖f a * ENNReal.toReal (pdf X ℙ a)‖₊ < ⊤ ** have : (fun x => ↑‖f x‖₊ * g x) =ᵐ[volume] fun x => ‖f x * (pdf X ℙ volume x).toReal‖₊ := by
refine' ae_eq_trans (Filter.EventuallyEq.mul (ae_eq_refl fun x => (‖f x‖₊ : ℝ≥0∞))
(ae_eq_trans hg.symm ofReal_toReal_ae_eq.symm)) _
simp_rw [← smul_eq_mul, nnnorm_smul, ENNReal.coe_mul, smul_eq_mul]
refine' Filter.EventuallyEq.mul (ae_eq_refl _) _
simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl] ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E inst✝ : IsFiniteMeasure ℙ X : Ω → ℝ f : ℝ → ℝ g : ℝ → ℝ≥0∞ hg : pdf X ℙ =ᶠ[ae volume] g hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤ this : (fun x => ↑‖f x‖₊ * g x) =ᶠ[ae volume] fun x => ↑‖f x * ENNReal.toReal (pdf X ℙ x)‖₊ ⊢ ∫⁻ (a : ℝ), ↑‖f a * ENNReal.toReal (pdf X ℙ a)‖₊ < ⊤ ** rwa [lt_top_iff_ne_top, ← lintegral_congr_ae this] ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E inst✝ : IsFiniteMeasure ℙ X : Ω → ℝ f : ℝ → ℝ g : ℝ → ℝ≥0∞ hg : pdf X ℙ =ᶠ[ae volume] g hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤ ⊢ (fun x => ↑‖f x‖₊ * g x) =ᶠ[ae volume] fun x => ↑‖f x * ENNReal.toReal (pdf X ℙ x)‖₊ ** refine' ae_eq_trans (Filter.EventuallyEq.mul (ae_eq_refl fun x => (‖f x‖₊ : ℝ≥0∞))
(ae_eq_trans hg.symm ofReal_toReal_ae_eq.symm)) _ ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E inst✝ : IsFiniteMeasure ℙ X : Ω → ℝ f : ℝ → ℝ g : ℝ → ℝ≥0∞ hg : pdf X ℙ =ᶠ[ae volume] g hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤ ⊢ (fun x => ↑‖f x‖₊ * ENNReal.ofReal (ENNReal.toReal (pdf X ℙ x))) =ᶠ[ae volume] fun x => ↑‖f x * ENNReal.toReal (pdf X ℙ x)‖₊ ** simp_rw [← smul_eq_mul, nnnorm_smul, ENNReal.coe_mul, smul_eq_mul] ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E inst✝ : IsFiniteMeasure ℙ X : Ω → ℝ f : ℝ → ℝ g : ℝ → ℝ≥0∞ hg : pdf X ℙ =ᶠ[ae volume] g hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤ ⊢ (fun x => ↑‖f x‖₊ * ENNReal.ofReal (ENNReal.toReal (pdf X ℙ x))) =ᶠ[ae volume] fun x => ↑‖f x‖₊ * ↑‖ENNReal.toReal (pdf X ℙ x)‖₊ ** refine' Filter.EventuallyEq.mul (ae_eq_refl _) _ ** Ω : Type u_1 E : Type u_2 inst✝¹ : MeasurableSpace E m : MeasurableSpace Ω ℙ : Measure Ω μ : Measure E inst✝ : IsFiniteMeasure ℙ X : Ω → ℝ f : ℝ → ℝ g : ℝ → ℝ≥0∞ hg : pdf X ℙ =ᶠ[ae volume] g hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤ ⊢ (fun x => ENNReal.ofReal (ENNReal.toReal (pdf X ℙ x))) =ᶠ[ae volume] fun x => ↑‖ENNReal.toReal (pdf X ℙ x)‖₊ ** simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl] ** Qed
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MeasureTheory.pdf.IsUniform.measure_preimage ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m✝ : MeasurableSpace Ω ℙ✝ : Measure Ω μ✝ : Measure E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : Measure E s : Set E hns : ↑↑μ s ≠ 0 hnt : ↑↑μ s ≠ ⊤ hms : MeasurableSet s hu : IsUniform X s ℙ A : Set E hA : MeasurableSet A ⊢ ↑↑ℙ (X ⁻¹' A) = ↑↑μ (s ∩ A) / ↑↑μ s ** haveI := hu.hasPDF hns hnt ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m✝ : MeasurableSpace Ω ℙ✝ : Measure Ω μ✝ : Measure E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : Measure E s : Set E hns : ↑↑μ s ≠ 0 hnt : ↑↑μ s ≠ ⊤ hms : MeasurableSet s hu : IsUniform X s ℙ A : Set E hA : MeasurableSet A this : HasPDF X ℙ ⊢ ↑↑ℙ (X ⁻¹' A) = ↑↑μ (s ∩ A) / ↑↑μ s ** rw [← Measure.map_apply (HasPDF.measurable X ℙ μ) hA, map_eq_set_lintegral_pdf X ℙ μ hA,
lintegral_congr_ae hu.restrict] ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m✝ : MeasurableSpace Ω ℙ✝ : Measure Ω μ✝ : Measure E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : Measure E s : Set E hns : ↑↑μ s ≠ 0 hnt : ↑↑μ s ≠ ⊤ hms : MeasurableSet s hu : IsUniform X s ℙ A : Set E hA : MeasurableSet A this : HasPDF X ℙ ⊢ ∫⁻ (a : E) in A, Set.indicator s ((↑↑μ s)⁻¹ • 1) a ∂μ = ↑↑μ (s ∩ A) / ↑↑μ s ** simp only [hms, hA, lintegral_indicator, Pi.smul_apply, Pi.one_apply, Algebra.id.smul_eq_mul,
mul_one, lintegral_const, restrict_apply', Set.univ_inter] ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m✝ : MeasurableSpace Ω ℙ✝ : Measure Ω μ✝ : Measure E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : Measure E s : Set E hns : ↑↑μ s ≠ 0 hnt : ↑↑μ s ≠ ⊤ hms : MeasurableSet s hu : IsUniform X s ℙ A : Set E hA : MeasurableSet A this : HasPDF X ℙ ⊢ (↑↑μ s)⁻¹ * ↑↑μ (s ∩ A) = ↑↑μ (s ∩ A) / ↑↑μ s ** rw [ENNReal.div_eq_inv_mul] ** Qed
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MeasureTheory.pdf.IsUniform.isProbabilityMeasure ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m✝ : MeasurableSpace Ω ℙ✝ : Measure Ω μ✝ : Measure E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : Measure E s : Set E hns : ↑↑μ s ≠ 0 hnt : ↑↑μ s ≠ ⊤ hms : MeasurableSet s hu : IsUniform X s ℙ ⊢ ↑↑ℙ Set.univ = 1 ** have : X ⁻¹' Set.univ = Set.univ := by simp only [Set.preimage_univ] ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m✝ : MeasurableSpace Ω ℙ✝ : Measure Ω μ✝ : Measure E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : Measure E s : Set E hns : ↑↑μ s ≠ 0 hnt : ↑↑μ s ≠ ⊤ hms : MeasurableSet s hu : IsUniform X s ℙ this : X ⁻¹' Set.univ = Set.univ ⊢ ↑↑ℙ Set.univ = 1 ** rw [← this, hu.measure_preimage hns hnt hms MeasurableSet.univ, Set.inter_univ,
ENNReal.div_self hns hnt] ** Ω : Type u_1 E : Type u_2 inst✝ : MeasurableSpace E m✝ : MeasurableSpace Ω ℙ✝ : Measure Ω μ✝ : Measure E m : MeasurableSpace Ω X : Ω → E ℙ : Measure Ω μ : Measure E s : Set E hns : ↑↑μ s ≠ 0 hnt : ↑↑μ s ≠ ⊤ hms : MeasurableSet s hu : IsUniform X s ℙ ⊢ X ⁻¹' Set.univ = Set.univ ** simp only [Set.preimage_univ] ** Qed
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MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁴ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝³ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝² : IsSFiniteKernel η a : α inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f : β × γ → E hf : AEStronglyMeasurable f (↑(κ ⊗ₖ η) a) ⊢ (fun x => ∫ (y : γ), f (x, y) ∂↑η (a, x)) =ᵐ[↑κ a] fun x => ∫ (y : γ), AEStronglyMeasurable.mk f hf (x, y) ∂↑η (a, x) ** filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with _ hx using integral_congr_ae hx ** Qed
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MeasureTheory.AEStronglyMeasurable.compProd_mk_left ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝³ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝² : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝¹ : IsSFiniteKernel η a : α δ : Type u_5 inst✝ : TopologicalSpace δ f : β × γ → δ hf : AEStronglyMeasurable f (↑(κ ⊗ₖ η) a) ⊢ ∀ᵐ (x : β) ∂↑κ a, AEStronglyMeasurable (fun y => f (x, y)) (↑η (a, x)) ** filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with x hx using
⟨fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prod_mk_left, hx⟩ ** Qed
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ProbabilityTheory.hasFiniteIntegral_compProd_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f ⊢ HasFiniteIntegral f ↔ (∀ᵐ (x : β) ∂↑κ a, HasFiniteIntegral fun y => f (x, y)) ∧ HasFiniteIntegral fun x => ∫ (y : γ), ‖f (x, y)‖ ∂↑η (a, x) ** simp only [HasFiniteIntegral] ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f ⊢ ∫⁻ (a : β × γ), ↑‖f a‖₊ ∂↑(κ ⊗ₖ η) a < ⊤ ↔ (∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤) ∧ ∫⁻ (a_1 : β), ↑‖∫ (y : γ), ‖f (a_1, y)‖ ∂↑η (a, a_1)‖₊ ∂↑κ a < ⊤ ** rw [kernel.lintegral_compProd _ _ _ h1f.ennnorm] ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f ⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b) ∂↑κ a < ⊤ ↔ (∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤) ∧ ∫⁻ (a_1 : β), ↑‖∫ (y : γ), ‖f (a_1, y)‖ ∂↑η (a, a_1)‖₊ ∂↑κ a < ⊤ ** have : ∀ x, ∀ᵐ y ∂η (a, x), 0 ≤ ‖f (x, y)‖ := fun x => eventually_of_forall fun y => norm_nonneg _ ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ ⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b) ∂↑κ a < ⊤ ↔ (∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤) ∧ ∫⁻ (a_1 : β), ↑‖∫ (y : γ), ‖f (a_1, y)‖ ∂↑η (a, a_1)‖₊ ∂↑κ a < ⊤ ** simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
(h1f.norm.comp_measurable measurable_prod_mk_left).aestronglyMeasurable,
ennnorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_coe_nnnorm] ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ ⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b) ∂↑κ a < ⊤ ↔ (∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤) ∧ ∫⁻ (a_1 : β), ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : γ), ↑‖f (a_1, a)‖₊ ∂↑η (a, a_1))) ∂↑κ a < ⊤ ** have : ∀ {p q r : Prop} (_ : r → p), (r ↔ p ∧ q) ↔ p → (r ↔ q) := fun {p q r} h1 => by
rw [← and_congr_right_iff, and_iff_right_of_imp h1] ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) ⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b) ∂↑κ a < ⊤ ↔ (∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤) ∧ ∫⁻ (a_1 : β), ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : γ), ↑‖f (a_1, a)‖₊ ∂↑η (a, a_1))) ∂↑κ a < ⊤ ** rw [this] ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ p q r : Prop h1 : r → p ⊢ (r ↔ p ∧ q) ↔ p → (r ↔ q) ** rw [← and_congr_right_iff, and_iff_right_of_imp h1] ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) ⊢ (∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤) → (∫⁻ (b : β), ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b) ∂↑κ a < ⊤ ↔ ∫⁻ (a_2 : β), ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : γ), ↑‖f (a_2, a)‖₊ ∂↑η (a, a_2))) ∂↑κ a < ⊤) ** intro h2f ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b) ∂↑κ a < ⊤ ↔ ∫⁻ (a_1 : β), ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : γ), ↑‖f (a_1, a)‖₊ ∂↑η (a, a_1))) ∂↑κ a < ⊤ ** rw [lintegral_congr_ae] ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ⊢ (fun b => ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b)) =ᵐ[↑κ a] fun a_1 => ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : γ), ↑‖f (a_1, a)‖₊ ∂↑η (a, a_1))) ** refine' h2f.mp _ ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ⊢ ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ → (fun b => ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b)) x = (fun a_2 => ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : γ), ↑‖f (a_2, a)‖₊ ∂↑η (a, a_2)))) x ** apply eventually_of_forall ** case hp α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ⊢ ∀ (x : β), ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ → (fun b => ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b)) x = (fun a_2 => ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : γ), ↑‖f (a_2, a)‖₊ ∂↑η (a, a_2)))) x ** intro x hx ** case hp α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ x : β hx : ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ⊢ (fun b => ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b)) x = (fun a_1 => ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : γ), ↑‖f (a_1, a)‖₊ ∂↑η (a, a_1)))) x ** dsimp only ** case hp α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ x : β hx : ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ⊢ ∫⁻ (c : γ), ↑‖f (x, c)‖₊ ∂↑η (a, x) = ENNReal.ofReal (ENNReal.toReal (∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x))) ** rw [ofReal_toReal] ** case hp α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ x : β hx : ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ⊢ ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) ≠ ⊤ ** rw [← lt_top_iff_ne_top] ** case hp α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ x : β hx : ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ⊢ ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ** exact hx ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) ⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b) ∂↑κ a < ⊤ → ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ** intro h2f ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∫⁻ (b : β), ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b) ∂↑κ a < ⊤ ⊢ ∀ᵐ (x : β) ∂↑κ a, ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) < ⊤ ** refine' ae_lt_top _ h2f.ne ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝² : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝¹ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝ : IsSFiniteKernel η a : α f : β × γ → E h1f : StronglyMeasurable f this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂↑η (a, x), 0 ≤ ‖f (x, y)‖ this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q)) h2f : ∫⁻ (b : β), ∫⁻ (c : γ), ↑‖f (b, c)‖₊ ∂↑η (a, b) ∂↑κ a < ⊤ ⊢ Measurable fun x => ∫⁻ (a : γ), ↑‖f (x, a)‖₊ ∂↑η (a, x) ** exact h1f.ennnorm.lintegral_kernel_prod_right'' ** Qed
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ProbabilityTheory.kernel.integral_fn_integral_add ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f g : β × γ → E F : E → E' hf : Integrable f hg : Integrable g ⊢ ∫ (x : β), F (∫ (y : γ), f (x, y) + g (x, y) ∂↑η (a, x)) ∂↑κ a = ∫ (x : β), F (∫ (y : γ), f (x, y) ∂↑η (a, x) + ∫ (y : γ), g (x, y) ∂↑η (a, x)) ∂↑κ a ** refine' integral_congr_ae _ ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f g : β × γ → E F : E → E' hf : Integrable f hg : Integrable g ⊢ (fun x => F (∫ (y : γ), f (x, y) + g (x, y) ∂↑η (a, x))) =ᵐ[↑κ a] fun x => F (∫ (y : γ), f (x, y) ∂↑η (a, x) + ∫ (y : γ), g (x, y) ∂↑η (a, x)) ** filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g ** case h α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f g : β × γ → E F : E → E' hf : Integrable f hg : Integrable g a✝ : β h2f : Integrable fun y => f (a✝, y) h2g : Integrable fun y => g (a✝, y) ⊢ F (∫ (y : γ), f (a✝, y) + g (a✝, y) ∂↑η (a, a✝)) = F (∫ (y : γ), f (a✝, y) ∂↑η (a, a✝) + ∫ (y : γ), g (a✝, y) ∂↑η (a, a✝)) ** simp [integral_add h2f h2g] ** Qed
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ProbabilityTheory.kernel.integral_fn_integral_sub ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f g : β × γ → E F : E → E' hf : Integrable f hg : Integrable g ⊢ ∫ (x : β), F (∫ (y : γ), f (x, y) - g (x, y) ∂↑η (a, x)) ∂↑κ a = ∫ (x : β), F (∫ (y : γ), f (x, y) ∂↑η (a, x) - ∫ (y : γ), g (x, y) ∂↑η (a, x)) ∂↑κ a ** refine' integral_congr_ae _ ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f g : β × γ → E F : E → E' hf : Integrable f hg : Integrable g ⊢ (fun x => F (∫ (y : γ), f (x, y) - g (x, y) ∂↑η (a, x))) =ᵐ[↑κ a] fun x => F (∫ (y : γ), f (x, y) ∂↑η (a, x) - ∫ (y : γ), g (x, y) ∂↑η (a, x)) ** filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g ** case h α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f g : β × γ → E F : E → E' hf : Integrable f hg : Integrable g a✝ : β h2f : Integrable fun y => f (a✝, y) h2g : Integrable fun y => g (a✝, y) ⊢ F (∫ (y : γ), f (a✝, y) - g (a✝, y) ∂↑η (a, a✝)) = F (∫ (y : γ), f (a✝, y) ∂↑η (a, a✝) - ∫ (y : γ), g (a✝, y) ∂↑η (a, a✝)) ** simp [integral_sub h2f h2g] ** Qed
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ProbabilityTheory.kernel.lintegral_fn_integral_sub ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f g : β × γ → E F : E → ℝ≥0∞ hf : Integrable f hg : Integrable g ⊢ ∫⁻ (x : β), F (∫ (y : γ), f (x, y) - g (x, y) ∂↑η (a, x)) ∂↑κ a = ∫⁻ (x : β), F (∫ (y : γ), f (x, y) ∂↑η (a, x) - ∫ (y : γ), g (x, y) ∂↑η (a, x)) ∂↑κ a ** refine' lintegral_congr_ae _ ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f g : β × γ → E F : E → ℝ≥0∞ hf : Integrable f hg : Integrable g ⊢ (fun x => F (∫ (y : γ), f (x, y) - g (x, y) ∂↑η (a, x))) =ᵐ[↑κ a] fun x => F (∫ (y : γ), f (x, y) ∂↑η (a, x) - ∫ (y : γ), g (x, y) ∂↑η (a, x)) ** filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g ** case h α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f g : β × γ → E F : E → ℝ≥0∞ hf : Integrable f hg : Integrable g a✝ : β h2f : Integrable fun y => f (a✝, y) h2g : Integrable fun y => g (a✝, y) ⊢ F (∫ (y : γ), f (a✝, y) - g (a✝, y) ∂↑η (a, a✝)) = F (∫ (y : γ), f (a✝, y) ∂↑η (a, a✝) - ∫ (y : γ), g (a✝, y) ∂↑η (a, a✝)) ** simp [integral_sub h2f h2g] ** Qed
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ProbabilityTheory.set_integral_compProd ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f : β × γ → E s : Set β t : Set γ hs : MeasurableSet s ht : MeasurableSet t hf : IntegrableOn f (s ×ˢ t) ⊢ ∫ (z : β × γ) in s ×ˢ t, f z ∂↑(κ ⊗ₖ η) a = ∫ (x : β) in s, ∫ (y : γ) in t, f (x, y) ∂↑η (a, x) ∂↑κ a ** rw [← kernel.restrict_apply (κ ⊗ₖ η) (hs.prod ht), ← compProd_restrict hs ht, integral_compProd] ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f : β × γ → E s : Set β t : Set γ hs : MeasurableSet s ht : MeasurableSet t hf : IntegrableOn f (s ×ˢ t) ⊢ ∫ (x : β), ∫ (y : γ), f (x, y) ∂↑(kernel.restrict η ht) (a, x) ∂↑(kernel.restrict κ hs) a = ∫ (x : β) in s, ∫ (y : γ) in t, f (x, y) ∂↑η (a, x) ∂↑κ a ** simp_rw [kernel.restrict_apply] ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f : β × γ → E s : Set β t : Set γ hs : MeasurableSet s ht : MeasurableSet t hf : IntegrableOn f (s ×ˢ t) ⊢ Integrable fun z => f z ** exact hf ** Qed
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ProbabilityTheory.set_integral_compProd_univ_right ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f : β × γ → E s : Set β hs : MeasurableSet s hf : IntegrableOn f (s ×ˢ univ) ⊢ ∫ (z : β × γ) in s ×ˢ univ, f z ∂↑(κ ⊗ₖ η) a = ∫ (x : β) in s, ∫ (y : γ), f (x, y) ∂↑η (a, x) ∂↑κ a ** simp_rw [set_integral_compProd hs MeasurableSet.univ hf, Measure.restrict_univ] ** Qed
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ProbabilityTheory.set_integral_compProd_univ_left ** α : Type u_1 β : Type u_2 γ : Type u_3 E : Type u_4 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ inst✝⁷ : NormedAddCommGroup E κ : { x // x ∈ kernel α β } inst✝⁶ : IsSFiniteKernel κ η : { x // x ∈ kernel (α × β) γ } inst✝⁵ : IsSFiniteKernel η a : α inst✝⁴ : NormedSpace ℝ E inst✝³ : CompleteSpace E E' : Type u_5 inst✝² : NormedAddCommGroup E' inst✝¹ : CompleteSpace E' inst✝ : NormedSpace ℝ E' f : β × γ → E t : Set γ ht : MeasurableSet t hf : IntegrableOn f (univ ×ˢ t) ⊢ ∫ (z : β × γ) in univ ×ˢ t, f z ∂↑(κ ⊗ₖ η) a = ∫ (x : β), ∫ (y : γ) in t, f (x, y) ∂↑η (a, x) ∂↑κ a ** simp_rw [set_integral_compProd MeasurableSet.univ ht hf, Measure.restrict_univ] ** Qed
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ProbabilityTheory.measurable_condexpKernel ** Ω : Type u_1 F : Type u_2 inst✝⁵ : TopologicalSpace Ω m mΩ : MeasurableSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω μ : Measure Ω inst✝¹ : IsFiniteMeasure μ inst✝ : NormedAddCommGroup F f : Ω → F s : Set Ω hs : MeasurableSet s ⊢ Measurable fun ω => ↑↑(↑(condexpKernel μ m) ω) s ** simp_rw [condexpKernel_apply_eq_condDistrib] ** Ω : Type u_1 F : Type u_2 inst✝⁵ : TopologicalSpace Ω m mΩ : MeasurableSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω μ : Measure Ω inst✝¹ : IsFiniteMeasure μ inst✝ : NormedAddCommGroup F f : Ω → F s : Set Ω hs : MeasurableSet s ⊢ Measurable fun ω => ↑↑(↑(condDistrib id id μ) (id ω)) s ** convert measurable_condDistrib (μ := μ) hs ** case h.e'_3 Ω : Type u_1 F : Type u_2 inst✝⁵ : TopologicalSpace Ω m mΩ : MeasurableSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω μ : Measure Ω inst✝¹ : IsFiniteMeasure μ inst✝ : NormedAddCommGroup F f : Ω → F s : Set Ω hs : MeasurableSet s ⊢ m ⊓ mΩ = MeasurableSpace.comap id (m ⊓ mΩ) ** rw [MeasurableSpace.comap_id] ** Qed
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ProbabilityTheory.integrable_toReal_condexpKernel ** Ω : Type u_1 F : Type u_2 inst✝⁵ : TopologicalSpace Ω m mΩ : MeasurableSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω μ : Measure Ω inst✝¹ : IsFiniteMeasure μ inst✝ : NormedAddCommGroup F f : Ω → F s : Set Ω hs : MeasurableSet s ⊢ Integrable fun ω => ENNReal.toReal (↑↑(↑(condexpKernel μ m) ω) s) ** rw [condexpKernel] ** Ω : Type u_1 F : Type u_2 inst✝⁵ : TopologicalSpace Ω m mΩ : MeasurableSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω μ : Measure Ω inst✝¹ : IsFiniteMeasure μ inst✝ : NormedAddCommGroup F f : Ω → F s : Set Ω hs : MeasurableSet s ⊢ Integrable fun ω => ENNReal.toReal (↑↑(↑(kernel.comap (condDistrib id id μ) id (_ : Measurable id)) ω) s) ** exact integrable_toReal_condDistrib (aemeasurable_id'' μ (inf_le_right : m ⊓ mΩ ≤ mΩ)) hs ** Qed
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ProbabilityTheory.kernel.ext_fun ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ η : { x // x ∈ kernel α β } h : ∀ (a : α) (f : β → ℝ≥0∞), Measurable f → ∫⁻ (b : β), f b ∂↑κ a = ∫⁻ (b : β), f b ∂↑η a ⊢ κ = η ** ext a s hs ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ η : { x // x ∈ kernel α β } h : ∀ (a : α) (f : β → ℝ≥0∞), Measurable f → ∫⁻ (b : β), f b ∂↑κ a = ∫⁻ (b : β), f b ∂↑η a a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(↑κ a) s = ↑↑(↑η a) s ** specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs) ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ η : { x // x ∈ kernel α β } a : α s : Set β hs : MeasurableSet s h : ∫⁻ (b : β), Set.indicator s (fun x => 1) b ∂↑κ a = ∫⁻ (b : β), Set.indicator s (fun x => 1) b ∂↑η a ⊢ ↑↑(↑κ a) s = ↑↑(↑η a) s ** simp_rw [lintegral_indicator_const hs, one_mul] at h ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ η : { x // x ∈ kernel α β } a : α s : Set β hs : MeasurableSet s h : ↑↑(↑κ a) s = ↑↑(↑η a) s ⊢ ↑↑(↑κ a) s = ↑↑(↑η a) s ** rw [h] ** Qed
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ProbabilityTheory.kernel.ext_fun_iff ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ η : { x // x ∈ kernel α β } h : κ = η a : α f : β → ℝ≥0∞ x✝ : Measurable f ⊢ ∫⁻ (b : β), f b ∂↑κ a = ∫⁻ (b : β), f b ∂↑η a ** rw [h] ** Qed
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ProbabilityTheory.kernel.sum_apply' ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ✝ : { x // x ∈ kernel α β } inst✝ : Countable ι κ : ι → { x // x ∈ kernel α β } a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(↑(kernel.sum κ) a) s = ∑' (n : ι), ↑↑(↑(κ n) a) s ** rw [sum_apply κ a, Measure.sum_apply _ hs] ** Qed
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ProbabilityTheory.kernel.sum_add ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ✝ : { x // x ∈ kernel α β } inst✝ : Countable ι κ η : ι → { x // x ∈ kernel α β } ⊢ (kernel.sum fun n => κ n + η n) = kernel.sum κ + kernel.sum η ** ext a s hs ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ✝ : { x // x ∈ kernel α β } inst✝ : Countable ι κ η : ι → { x // x ∈ kernel α β } a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(↑(kernel.sum fun n => κ n + η n) a) s = ↑↑(↑(kernel.sum κ + kernel.sum η) a) s ** simp only [coeFn_add, Pi.add_apply, sum_apply, Measure.sum_apply _ hs, Pi.add_apply,
Measure.coe_add, tsum_add ENNReal.summable ENNReal.summable] ** Qed
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ProbabilityTheory.kernel.measure_sum_seq ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ✝ κ : { x // x ∈ kernel α β } h : IsSFiniteKernel κ a : α ⊢ (Measure.sum fun n => ↑(seq κ n) a) = ↑κ a ** rw [← kernel.sum_apply, kernel_sum_seq κ] ** Qed
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ProbabilityTheory.kernel.isSFiniteKernel_sum_of_denumerable ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } inst✝ : Denumerable ι κs : ι → { x // x ∈ kernel α β } hκs : ∀ (n : ι), IsSFiniteKernel (κs n) ⊢ IsSFiniteKernel (kernel.sum κs) ** let e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } inst✝ : Denumerable ι κs : ι → { x // x ∈ kernel α β } hκs : ∀ (n : ι), IsSFiniteKernel (κs n) e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm ⊢ IsSFiniteKernel (kernel.sum κs) ** refine' ⟨⟨fun n => seq (κs (e n).1) (e n).2, inferInstance, _⟩⟩ ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } inst✝ : Denumerable ι κs : ι → { x // x ∈ kernel α β } hκs : ∀ (n : ι), IsSFiniteKernel (κs n) e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm ⊢ kernel.sum κs = kernel.sum fun n => seq (κs (↑e n).1) (↑e n).2 ** have hκ_eq : kernel.sum κs = kernel.sum fun n => kernel.sum (seq (κs n)) := by
simp_rw [kernel_sum_seq] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } inst✝ : Denumerable ι κs : ι → { x // x ∈ kernel α β } hκs : ∀ (n : ι), IsSFiniteKernel (κs n) e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm hκ_eq : kernel.sum κs = kernel.sum fun n => kernel.sum (seq (κs n)) ⊢ kernel.sum κs = kernel.sum fun n => seq (κs (↑e n).1) (↑e n).2 ** ext a s hs ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } inst✝ : Denumerable ι κs : ι → { x // x ∈ kernel α β } hκs : ∀ (n : ι), IsSFiniteKernel (κs n) e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm hκ_eq : kernel.sum κs = kernel.sum fun n => kernel.sum (seq (κs n)) a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(↑(kernel.sum κs) a) s = ↑↑(↑(kernel.sum fun n => seq (κs (↑e n).1) (↑e n).2) a) s ** rw [hκ_eq] ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } inst✝ : Denumerable ι κs : ι → { x // x ∈ kernel α β } hκs : ∀ (n : ι), IsSFiniteKernel (κs n) e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm hκ_eq : kernel.sum κs = kernel.sum fun n => kernel.sum (seq (κs n)) a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(↑(kernel.sum fun n => kernel.sum (seq (κs n))) a) s = ↑↑(↑(kernel.sum fun n => seq (κs (↑e n).1) (↑e n).2) a) s ** simp_rw [kernel.sum_apply' _ _ hs] ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } inst✝ : Denumerable ι κs : ι → { x // x ∈ kernel α β } hκs : ∀ (n : ι), IsSFiniteKernel (κs n) e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm hκ_eq : kernel.sum κs = kernel.sum fun n => kernel.sum (seq (κs n)) a : α s : Set β hs : MeasurableSet s ⊢ ∑' (n : ι) (n_1 : ℕ), ↑↑(↑(seq (κs n) n_1) a) s = ∑' (n : ℕ), ↑↑(↑(seq (κs (↑(Denumerable.eqv (ι × ℕ)).symm n).1) (↑(Denumerable.eqv (ι × ℕ)).symm n).2) a) s ** change (∑' i, ∑' m, seq (κs i) m a s) = ∑' n, (fun im : ι × ℕ => seq (κs im.fst) im.snd a s) (e n) ** case h.h α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } inst✝ : Denumerable ι κs : ι → { x // x ∈ kernel α β } hκs : ∀ (n : ι), IsSFiniteKernel (κs n) e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm hκ_eq : kernel.sum κs = kernel.sum fun n => kernel.sum (seq (κs n)) a : α s : Set β hs : MeasurableSet s ⊢ ∑' (i : ι) (m : ℕ), ↑↑(↑(seq (κs i) m) a) s = ∑' (n : ℕ), (fun im => ↑↑(↑(seq (κs im.1) im.2) a) s) (↑e n) ** rw [e.tsum_eq (fun im : ι × ℕ => seq (κs im.fst) im.snd a s),
tsum_prod' ENNReal.summable fun _ => ENNReal.summable] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } inst✝ : Denumerable ι κs : ι → { x // x ∈ kernel α β } hκs : ∀ (n : ι), IsSFiniteKernel (κs n) e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm ⊢ kernel.sum κs = kernel.sum fun n => kernel.sum (seq (κs n)) ** simp_rw [kernel_sum_seq] ** Qed
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ProbabilityTheory.kernel.deterministic_apply' ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } f : α → β hf : Measurable f a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(↑(deterministic f hf) a) s = Set.indicator s (fun x => 1) (f a) ** rw [deterministic] ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } f : α → β hf : Measurable f a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(↑{ val := fun a => Measure.dirac (f a), property := (_ : Measurable fun a => Measure.dirac (f a)) } a) s = Set.indicator s (fun x => 1) (f a) ** change Measure.dirac (f a) s = s.indicator 1 (f a) ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } f : α → β hf : Measurable f a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(Measure.dirac (f a)) s = Set.indicator s 1 (f a) ** simp_rw [Measure.dirac_apply' _ hs] ** Qed
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ProbabilityTheory.kernel.lintegral_deterministic' ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } f : β → ℝ≥0∞ g : α → β a : α hg : Measurable g hf : Measurable f ⊢ ∫⁻ (x : β), f x ∂↑(deterministic g hg) a = f (g a) ** rw [kernel.deterministic_apply, lintegral_dirac' _ hf] ** Qed
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ProbabilityTheory.kernel.lintegral_deterministic ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } f : β → ℝ≥0∞ g : α → β a : α hg : Measurable g inst✝ : MeasurableSingletonClass β ⊢ ∫⁻ (x : β), f x ∂↑(deterministic g hg) a = f (g a) ** rw [kernel.deterministic_apply, lintegral_dirac (g a) f] ** Qed
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ProbabilityTheory.kernel.integral_deterministic' ** α : Type u_1 β : Type u_2 ι : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β κ : { x // x ∈ kernel α β } E : Type u_4 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f : β → E g : α → β a : α hg : Measurable g hf : StronglyMeasurable f ⊢ ∫ (x : β), f x ∂↑(deterministic g hg) a = f (g a) ** rw [kernel.deterministic_apply, integral_dirac' _ _ hf] ** Qed
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