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ProbabilityTheory.measure_ge_le_exp_cgf ** Ω : Type u_1 ι : Type u_2 m : MeasurableSpace Ω X : Ω → ℝ p : ℕ μ : Measure Ω t : ℝ inst✝ : IsFiniteMeasure μ ε : ℝ ht : 0 ≤ t h_int : Integrable fun ω => rexp (t * X ω) ⊢ ENNReal.toReal (↑↑μ {ω | ε ≤ X ω}) ≤ rexp (-t * ε + cgf X μ t) ** refine' (measure_ge_le_exp_mul_mgf ε ht h_int).trans _ ** Ω : Type u_1 ι : Type u_2 m : MeasurableSpace Ω X : Ω → ℝ p : ℕ μ : Measure Ω t : ℝ inst✝ : IsFiniteMeasure μ ε : ℝ ht : 0 ≤ t h_int : Integrable fun ω => rexp (t * X ω) ⊢ rexp (-t * ε) * mgf (fun ω => X ω) μ t ≤ rexp (-t * ε + cgf X μ t) ** rw [exp_add] ** Ω : Type u_1 ι : Type u_2 m : MeasurableSpace Ω X : Ω → ℝ p : ℕ μ : Measure Ω t : ℝ inst✝ : IsFiniteMeasure μ ε : ℝ ht : 0 ≤ t h_int : Integrable fun ω => rexp (t * X ω) ⊢ rexp (-t * ε) * mgf (fun ω => X ω) μ t ≤ rexp (-t * ε) * rexp (cgf X μ t) ** exact mul_le_mul le_rfl (le_exp_log _) mgf_nonneg (exp_pos _).le ** Qed
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ProbabilityTheory.iIndepFun.indep_comap_natural_of_lt ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : IsProbabilityMeasure μ ι : Type u_2 β : Type u_3 inst✝² : LinearOrder ι mβ : MeasurableSpace β inst✝¹ : NormedAddCommGroup β inst✝ : BorelSpace β f : ι → Ω → β i j : ι s : ι → Set Ω hf : ∀ (i : ι), StronglyMeasurable (f i) hfi : iIndepFun (fun x => mβ) f hij : i < j ⊢ Indep (MeasurableSpace.comap (f j) mβ) (↑(Filtration.natural f hf) i) ** suffices Indep (⨆ k ∈ ({j} : Set ι), MeasurableSpace.comap (f k) mβ)
(⨆ k ∈ {k | k ≤ i}, MeasurableSpace.comap (f k) mβ) μ by rwa [iSup_singleton] at this ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : IsProbabilityMeasure μ ι : Type u_2 β : Type u_3 inst✝² : LinearOrder ι mβ : MeasurableSpace β inst✝¹ : NormedAddCommGroup β inst✝ : BorelSpace β f : ι → Ω → β i j : ι s : ι → Set Ω hf : ∀ (i : ι), StronglyMeasurable (f i) hfi : iIndepFun (fun x => mβ) f hij : i < j ⊢ Indep (⨆ k ∈ {j}, MeasurableSpace.comap (f k) mβ) (⨆ k ∈ {k | k ≤ i}, MeasurableSpace.comap (f k) mβ) ** exact indep_iSup_of_disjoint (fun k => (hf k).measurable.comap_le) hfi (by simpa) ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : IsProbabilityMeasure μ ι : Type u_2 β : Type u_3 inst✝² : LinearOrder ι mβ : MeasurableSpace β inst✝¹ : NormedAddCommGroup β inst✝ : BorelSpace β f : ι → Ω → β i j : ι s : ι → Set Ω hf : ∀ (i : ι), StronglyMeasurable (f i) hfi : iIndepFun (fun x => mβ) f hij : i < j this : Indep (⨆ k ∈ {j}, MeasurableSpace.comap (f k) mβ) (⨆ k ∈ {k | k ≤ i}, MeasurableSpace.comap (f k) mβ) ⊢ Indep (MeasurableSpace.comap (f j) mβ) (↑(Filtration.natural f hf) i) ** rwa [iSup_singleton] at this ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : IsProbabilityMeasure μ ι : Type u_2 β : Type u_3 inst✝² : LinearOrder ι mβ : MeasurableSpace β inst✝¹ : NormedAddCommGroup β inst✝ : BorelSpace β f : ι → Ω → β i j : ι s : ι → Set Ω hf : ∀ (i : ι), StronglyMeasurable (f i) hfi : iIndepFun (fun x => mβ) f hij : i < j ⊢ Disjoint {j} {k | k ≤ i} ** simpa ** Qed
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MeasureTheory.Submartingale.upcrossings_ae_lt_top ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R ⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (a b : ℚ), a < b → upcrossings (↑a) (↑b) f ω < ⊤ ** simp only [ae_all_iff, eventually_imp_distrib_left] ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R ⊢ ∀ (i i_1 : ℚ), i < i_1 → ∀ᵐ (x : Ω) ∂μ, upcrossings (↑i) (↑i_1) f x < ⊤ ** rintro a b hab ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a✝ b✝ : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R a b : ℚ hab : a < b ⊢ ∀ᵐ (x : Ω) ∂μ, upcrossings (↑a) (↑b) f x < ⊤ ** exact hf.upcrossings_ae_lt_top' hbdd (Rat.cast_lt.2 hab) ** Qed
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MeasureTheory.Submartingale.ae_tendsto_limitProcess ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R ⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (limitProcess f ℱ μ ω)) ** classical
suffices
∃ g, StronglyMeasurable[⨆ n, ℱ n] g ∧ ∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) by
rw [limitProcess, dif_pos this]
exact (Classical.choose_spec this).2
set g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then h.choose else 0
have hle : ⨆ n, ℱ n ≤ m0 := sSup_le fun m ⟨n, hn⟩ => hn ▸ ℱ.le _
have hg' : ∀ᵐ ω ∂μ.trim hle, Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) := by
filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with ω hω
simp_rw [dif_pos hω]
exact hω.choose_spec
have hg'm : @AEStronglyMeasurable _ _ _ (⨆ n, ℱ n) g' (μ.trim hle) :=
(@aemeasurable_of_tendsto_metrizable_ae' _ _ (⨆ n, ℱ n) _ _ _ _ _ _ _
(fun n => ((hf.stronglyMeasurable n).measurable.mono (le_sSup ⟨n, rfl⟩ : ℱ n ≤ ⨆ n, ℱ n)
le_rfl).aemeasurable) hg').aestronglyMeasurable
obtain ⟨g, hgm, hae⟩ := hg'm
have hg : ∀ᵐ ω ∂μ.trim hle, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) := by
filter_upwards [hae, hg'] with ω hω hg'ω
exact hω ▸ hg'ω
exact ⟨g, hgm, measure_eq_zero_of_trim_eq_zero hle hg⟩ ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R ⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (limitProcess f ℱ μ ω)) ** suffices
∃ g, StronglyMeasurable[⨆ n, ℱ n] g ∧ ∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) by
rw [limitProcess, dif_pos this]
exact (Classical.choose_spec this).2 ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R ⊢ ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ** set g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then h.choose else 0 ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 ⊢ ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ** have hle : ⨆ n, ℱ n ≤ m0 := sSup_le fun m ⟨n, hn⟩ => hn ▸ ℱ.le _ ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 ⊢ ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ** have hg' : ∀ᵐ ω ∂μ.trim hle, Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) := by
filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with ω hω
simp_rw [dif_pos hω]
exact hω.choose_spec ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 hg' : ∀ᵐ (ω : Ω) ∂Measure.trim μ hle, Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) ⊢ ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ** have hg'm : @AEStronglyMeasurable _ _ _ (⨆ n, ℱ n) g' (μ.trim hle) :=
(@aemeasurable_of_tendsto_metrizable_ae' _ _ (⨆ n, ℱ n) _ _ _ _ _ _ _
(fun n => ((hf.stronglyMeasurable n).measurable.mono (le_sSup ⟨n, rfl⟩ : ℱ n ≤ ⨆ n, ℱ n)
le_rfl).aemeasurable) hg').aestronglyMeasurable ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 hg' : ∀ᵐ (ω : Ω) ∂Measure.trim μ hle, Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) hg'm : AEStronglyMeasurable g' (Measure.trim μ hle) ⊢ ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ** obtain ⟨g, hgm, hae⟩ := hg'm ** case intro.intro Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 hg' : ∀ᵐ (ω : Ω) ∂Measure.trim μ hle, Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) g : Ω → ℝ hgm : StronglyMeasurable g hae : g' =ᵐ[Measure.trim μ hle] g ⊢ ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ** have hg : ∀ᵐ ω ∂μ.trim hle, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) := by
filter_upwards [hae, hg'] with ω hω hg'ω
exact hω ▸ hg'ω ** case intro.intro Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 hg' : ∀ᵐ (ω : Ω) ∂Measure.trim μ hle, Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) g : Ω → ℝ hgm : StronglyMeasurable g hae : g' =ᵐ[Measure.trim μ hle] g hg : ∀ᵐ (ω : Ω) ∂Measure.trim μ hle, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ⊢ ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ** exact ⟨g, hgm, measure_eq_zero_of_trim_eq_zero hle hg⟩ ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R this : ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (limitProcess f ℱ μ ω)) ** rw [limitProcess, dif_pos this] ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R this : ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (Classical.choose this ω)) ** exact (Classical.choose_spec this).2 ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 ⊢ ∀ᵐ (ω : Ω) ∂Measure.trim μ hle, Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) ** filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with ω hω ** case h Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω✝ : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 ω : Ω hω : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) ⊢ Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) ** simp_rw [dif_pos hω] ** case h Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω✝ : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 ω : Ω hω : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) ⊢ Tendsto (fun n => f n ω) atTop (𝓝 (Exists.choose hω)) ** exact hω.choose_spec ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 hg' : ∀ᵐ (ω : Ω) ∂Measure.trim μ hle, Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) g : Ω → ℝ hgm : StronglyMeasurable g hae : g' =ᵐ[Measure.trim μ hle] g ⊢ ∀ᵐ (ω : Ω) ∂Measure.trim μ hle, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ** filter_upwards [hae, hg'] with ω hω hg'ω ** case h Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω✝ : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hbdd : ∀ (n : ℕ), snorm (f n) 1 μ ≤ ↑R g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then Exists.choose h else 0 hle : ⨆ n, ↑ℱ n ≤ m0 hg' : ∀ᵐ (ω : Ω) ∂Measure.trim μ hle, Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) g : Ω → ℝ hgm : StronglyMeasurable g hae : g' =ᵐ[Measure.trim μ hle] g ω : Ω hω : g' ω = g ω hg'ω : Tendsto (fun n => f n ω) atTop (𝓝 (g' ω)) ⊢ Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) ** exact hω ▸ hg'ω ** Qed
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MeasureTheory.Submartingale.tendsto_snorm_one_limitProcess ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g : Ω → ℝ hf : Submartingale f ℱ μ hunif : UniformIntegrable f 1 μ ⊢ Tendsto (fun n => snorm (f n - limitProcess f ℱ μ) 1 μ) atTop (𝓝 0) ** obtain ⟨R, hR⟩ := hunif.2.2 ** case intro Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R✝ : ℝ≥0 inst✝ : IsFiniteMeasure μ g : Ω → ℝ hf : Submartingale f ℱ μ hunif : UniformIntegrable f 1 μ R : ℝ≥0 hR : ∀ (i : ℕ), snorm (f i) 1 μ ≤ ↑R ⊢ Tendsto (fun n => snorm (f n - limitProcess f ℱ μ) 1 μ) atTop (𝓝 0) ** have hmeas : ∀ n, AEStronglyMeasurable (f n) μ := fun n =>
((hf.stronglyMeasurable n).mono (ℱ.le _)).aestronglyMeasurable ** case intro Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R✝ : ℝ≥0 inst✝ : IsFiniteMeasure μ g : Ω → ℝ hf : Submartingale f ℱ μ hunif : UniformIntegrable f 1 μ R : ℝ≥0 hR : ∀ (i : ℕ), snorm (f i) 1 μ ≤ ↑R hmeas : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ ⊢ Tendsto (fun n => snorm (f n - limitProcess f ℱ μ) 1 μ) atTop (𝓝 0) ** exact tendsto_Lp_of_tendstoInMeasure _ le_rfl ENNReal.one_ne_top hmeas
(memℒp_limitProcess_of_snorm_bdd hmeas hR) hunif.2.1
(tendstoInMeasure_of_tendsto_ae hmeas <| hf.ae_tendsto_limitProcess hR) ** Qed
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MeasureTheory.Martingale.eq_condexp_of_tendsto_snorm ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ✝ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ✝ g : Ω → ℝ μ : Measure Ω hf : Martingale f ℱ μ hg : Integrable g hgtends : Tendsto (fun n => snorm (f n - g) 1 μ) atTop (𝓝 0) n : ℕ ⊢ f n =ᵐ[μ] μ[g|↑ℱ n] ** rw [← sub_ae_eq_zero, ← snorm_eq_zero_iff (((hf.stronglyMeasurable n).mono (ℱ.le _)).sub
(stronglyMeasurable_condexp.mono (ℱ.le _))).aestronglyMeasurable one_ne_zero] ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ✝ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ✝ g : Ω → ℝ μ : Measure Ω hf : Martingale f ℱ μ hg : Integrable g hgtends : Tendsto (fun n => snorm (f n - g) 1 μ) atTop (𝓝 0) n : ℕ ⊢ snorm (f n - μ[g|↑ℱ n]) 1 μ = 0 ** have ht : Tendsto (fun m => snorm (μ[f m - g|ℱ n]) 1 μ) atTop (𝓝 0) :=
haveI hint : ∀ m, Integrable (f m - g) μ := fun m => (hf.integrable m).sub hg
tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hgtends (fun m => zero_le _)
fun m => snorm_one_condexp_le_snorm _ ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ✝ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ✝ g : Ω → ℝ μ : Measure Ω hf : Martingale f ℱ μ hg : Integrable g hgtends : Tendsto (fun n => snorm (f n - g) 1 μ) atTop (𝓝 0) n : ℕ ht : Tendsto (fun m => snorm (μ[f m - g|↑ℱ n]) 1 μ) atTop (𝓝 0) ⊢ snorm (f n - μ[g|↑ℱ n]) 1 μ = 0 ** have hev : ∀ m ≥ n, snorm (μ[f m - g|ℱ n]) 1 μ = snorm (f n - μ[g|ℱ n]) 1 μ := by
refine' fun m hm => snorm_congr_ae ((condexp_sub (hf.integrable m) hg).trans _)
filter_upwards [hf.2 n m hm] with x hx
simp only [hx, Pi.sub_apply] ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ✝ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ✝ g : Ω → ℝ μ : Measure Ω hf : Martingale f ℱ μ hg : Integrable g hgtends : Tendsto (fun n => snorm (f n - g) 1 μ) atTop (𝓝 0) n : ℕ ht : Tendsto (fun m => snorm (μ[f m - g|↑ℱ n]) 1 μ) atTop (𝓝 0) hev : ∀ (m : ℕ), m ≥ n → snorm (μ[f m - g|↑ℱ n]) 1 μ = snorm (f n - μ[g|↑ℱ n]) 1 μ ⊢ snorm (f n - μ[g|↑ℱ n]) 1 μ = 0 ** exact tendsto_nhds_unique (tendsto_atTop_of_eventually_const hev) ht ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ✝ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ✝ g : Ω → ℝ μ : Measure Ω hf : Martingale f ℱ μ hg : Integrable g hgtends : Tendsto (fun n => snorm (f n - g) 1 μ) atTop (𝓝 0) n : ℕ ht : Tendsto (fun m => snorm (μ[f m - g|↑ℱ n]) 1 μ) atTop (𝓝 0) ⊢ ∀ (m : ℕ), m ≥ n → snorm (μ[f m - g|↑ℱ n]) 1 μ = snorm (f n - μ[g|↑ℱ n]) 1 μ ** refine' fun m hm => snorm_congr_ae ((condexp_sub (hf.integrable m) hg).trans _) ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ✝ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ✝ g : Ω → ℝ μ : Measure Ω hf : Martingale f ℱ μ hg : Integrable g hgtends : Tendsto (fun n => snorm (f n - g) 1 μ) atTop (𝓝 0) n : ℕ ht : Tendsto (fun m => snorm (μ[f m - g|↑ℱ n]) 1 μ) atTop (𝓝 0) m : ℕ hm : m ≥ n ⊢ μ[f m|↑ℱ n] - μ[g|↑ℱ n] =ᵐ[μ] f n - μ[g|↑ℱ n] ** filter_upwards [hf.2 n m hm] with x hx ** case h Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ✝ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ✝ g : Ω → ℝ μ : Measure Ω hf : Martingale f ℱ μ hg : Integrable g hgtends : Tendsto (fun n => snorm (f n - g) 1 μ) atTop (𝓝 0) n : ℕ ht : Tendsto (fun m => snorm (μ[f m - g|↑ℱ n]) 1 μ) atTop (𝓝 0) m : ℕ hm : m ≥ n x : Ω hx : (μ[f m|↑ℱ n]) x = f n x ⊢ (μ[f m|↑ℱ n] - μ[g|↑ℱ n]) x = (f n - μ[g|↑ℱ n]) x ** simp only [hx, Pi.sub_apply] ** Qed
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MeasureTheory.tendsto_ae_condexp ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ⊢ ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) ** have ht : ∀ᵐ x ∂μ, Tendsto (fun n => (μ[μ[g|⨆ n, ℱ n]|ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ℱ n]) x)) :=
integrable_condexp.tendsto_ae_condexp stronglyMeasurable_condexp ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ht : ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) ⊢ ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) ** have heq : ∀ n, ∀ᵐ x ∂μ, (μ[μ[g|⨆ n, ℱ n]|ℱ n]) x = (μ[g|ℱ n]) x := fun n =>
condexp_condexp_of_le (le_iSup _ n) (iSup_le fun n => ℱ.le n) ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ht : ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) heq : ∀ (n : ℕ), ∀ᵐ (x : Ω) ∂μ, (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x = (μ[g|↑ℱ n]) x ⊢ ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) ** rw [← ae_all_iff] at heq ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ht : ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) heq : ∀ᵐ (a : Ω) ∂μ, ∀ (i : ℕ), (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ i]) a = (μ[g|↑ℱ i]) a ⊢ ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) ** filter_upwards [heq, ht] with x hxeq hxt ** case h Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ht : ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) heq : ∀ᵐ (a : Ω) ∂μ, ∀ (i : ℕ), (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ i]) a = (μ[g|↑ℱ i]) a x : Ω hxeq : ∀ (i : ℕ), (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ i]) x = (μ[g|↑ℱ i]) x hxt : Tendsto (fun n => (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) ⊢ Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x)) ** exact hxt.congr hxeq ** Qed
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MeasureTheory.tendsto_snorm_condexp ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ⊢ Tendsto (fun n => snorm (μ[g|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) 1 μ) atTop (𝓝 0) ** have ht : Tendsto (fun n => snorm (μ[μ[g|⨆ n, ℱ n]|ℱ n] - μ[g|⨆ n, ℱ n]) 1 μ) atTop (𝓝 0) :=
integrable_condexp.tendsto_snorm_condexp stronglyMeasurable_condexp ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ht : Tendsto (fun n => snorm (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) 1 μ) atTop (𝓝 0) ⊢ Tendsto (fun n => snorm (μ[g|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) 1 μ) atTop (𝓝 0) ** have heq : ∀ n, ∀ᵐ x ∂μ, (μ[μ[g|⨆ n, ℱ n]|ℱ n]) x = (μ[g|ℱ n]) x := fun n =>
condexp_condexp_of_le (le_iSup _ n) (iSup_le fun n => ℱ.le n) ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ht : Tendsto (fun n => snorm (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) 1 μ) atTop (𝓝 0) heq : ∀ (n : ℕ), ∀ᵐ (x : Ω) ∂μ, (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x = (μ[g|↑ℱ n]) x ⊢ Tendsto (fun n => snorm (μ[g|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) 1 μ) atTop (𝓝 0) ** refine' ht.congr fun n => snorm_congr_ae _ ** Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ht : Tendsto (fun n => snorm (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) 1 μ) atTop (𝓝 0) heq : ∀ (n : ℕ), ∀ᵐ (x : Ω) ∂μ, (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x = (μ[g|↑ℱ n]) x n : ℕ ⊢ μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n] - μ[g|⨆ n, ↑ℱ n] =ᵐ[μ] μ[g|↑ℱ n] - μ[g|⨆ n, ↑ℱ n] ** filter_upwards [heq n] with x hxeq ** case h Ω : Type u_1 ι : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 a b : ℝ f : ℕ → Ω → ℝ ω : Ω R : ℝ≥0 inst✝ : IsFiniteMeasure μ g✝ g : Ω → ℝ ht : Tendsto (fun n => snorm (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) 1 μ) atTop (𝓝 0) heq : ∀ (n : ℕ), ∀ᵐ (x : Ω) ∂μ, (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x = (μ[g|↑ℱ n]) x n : ℕ x : Ω hxeq : (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n]) x = (μ[g|↑ℱ n]) x ⊢ (μ[μ[g|⨆ n, ↑ℱ n]|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) x = (μ[g|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) x ** simp only [hxeq, Pi.sub_apply] ** Qed
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MeasureTheory.Submartingale.stoppedValue_leastGE ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ ⊢ Submartingale (fun i => stoppedValue f (leastGE f r i)) ℱ μ ** rw [submartingale_iff_expected_stoppedValue_mono] ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ ⊢ ∀ (τ π : Ω → ℕ), IsStoppingTime ℱ τ → IsStoppingTime ℱ π → τ ≤ π → (∃ N, ∀ (x : Ω), π x ≤ N) → ∫ (x : Ω), stoppedValue (fun i => stoppedValue f (leastGE f r i)) τ x ∂μ ≤ ∫ (x : Ω), stoppedValue (fun i => stoppedValue f (leastGE f r i)) π x ∂μ ** intro σ π hσ hπ hσ_le_π hπ_bdd ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ σ π : Ω → ℕ hσ : IsStoppingTime ℱ σ hπ : IsStoppingTime ℱ π hσ_le_π : σ ≤ π hπ_bdd : ∃ N, ∀ (x : Ω), π x ≤ N ⊢ ∫ (x : Ω), stoppedValue (fun i => stoppedValue f (leastGE f r i)) σ x ∂μ ≤ ∫ (x : Ω), stoppedValue (fun i => stoppedValue f (leastGE f r i)) π x ∂μ ** obtain ⟨n, hπ_le_n⟩ := hπ_bdd ** case intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ σ π : Ω → ℕ hσ : IsStoppingTime ℱ σ hπ : IsStoppingTime ℱ π hσ_le_π : σ ≤ π n : ℕ hπ_le_n : ∀ (x : Ω), π x ≤ n ⊢ ∫ (x : Ω), stoppedValue (fun i => stoppedValue f (leastGE f r i)) σ x ∂μ ≤ ∫ (x : Ω), stoppedValue (fun i => stoppedValue f (leastGE f r i)) π x ∂μ ** simp_rw [stoppedValue_stoppedValue_leastGE f σ r fun i => (hσ_le_π i).trans (hπ_le_n i)] ** case intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ σ π : Ω → ℕ hσ : IsStoppingTime ℱ σ hπ : IsStoppingTime ℱ π hσ_le_π : σ ≤ π n : ℕ hπ_le_n : ∀ (x : Ω), π x ≤ n ⊢ ∫ (x : Ω), stoppedValue (stoppedProcess f (leastGE f r n)) σ x ∂μ ≤ ∫ (x : Ω), stoppedValue (fun i => stoppedValue f (leastGE f r i)) π x ∂μ ** simp_rw [stoppedValue_stoppedValue_leastGE f π r hπ_le_n] ** case intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ σ π : Ω → ℕ hσ : IsStoppingTime ℱ σ hπ : IsStoppingTime ℱ π hσ_le_π : σ ≤ π n : ℕ hπ_le_n : ∀ (x : Ω), π x ≤ n ⊢ ∫ (x : Ω), stoppedValue (stoppedProcess f (leastGE f r n)) σ x ∂μ ≤ ∫ (x : Ω), stoppedValue (stoppedProcess f (leastGE f r n)) π x ∂μ ** refine' hf.expected_stoppedValue_mono _ _ _ fun ω => (min_le_left _ _).trans (hπ_le_n ω) ** case intro.refine'_1 Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ σ π : Ω → ℕ hσ : IsStoppingTime ℱ σ hπ : IsStoppingTime ℱ π hσ_le_π : σ ≤ π n : ℕ hπ_le_n : ∀ (x : Ω), π x ≤ n ⊢ IsStoppingTime ℱ fun x => min (σ x) (leastGE f r n x) ** exact hσ.min (hf.adapted.isStoppingTime_leastGE _ _) ** case intro.refine'_2 Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ σ π : Ω → ℕ hσ : IsStoppingTime ℱ σ hπ : IsStoppingTime ℱ π hσ_le_π : σ ≤ π n : ℕ hπ_le_n : ∀ (x : Ω), π x ≤ n ⊢ IsStoppingTime ℱ fun x => min (π x) (leastGE f r n x) ** exact hπ.min (hf.adapted.isStoppingTime_leastGE _ _) ** case intro.refine'_3 Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ σ π : Ω → ℕ hσ : IsStoppingTime ℱ σ hπ : IsStoppingTime ℱ π hσ_le_π : σ ≤ π n : ℕ hπ_le_n : ∀ (x : Ω), π x ≤ n ⊢ (fun x => min (σ x) (leastGE f r n x)) ≤ fun x => min (π x) (leastGE f r n x) ** exact fun ω => min_le_min (hσ_le_π ω) le_rfl ** case hadp Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ ⊢ Adapted ℱ fun i => stoppedValue f (leastGE f r i) ** exact fun i => stronglyMeasurable_stoppedValue_of_le hf.adapted.progMeasurable_of_discrete
(hf.adapted.isStoppingTime_leastGE _ _) leastGE_le ** case hint Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ r : ℝ ⊢ ∀ (i : ℕ), Integrable (stoppedValue f (leastGE f r i)) ** exact fun i => integrable_stoppedValue _ (hf.adapted.isStoppingTime_leastGE _ _) hf.integrable
leastGE_le ** Qed
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MeasureTheory.Submartingale.stoppedValue_leastGE_snorm_le' ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hr : 0 ≤ r hf0 : f 0 = 0 hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R i : ℕ ⊢ snorm (stoppedValue f (leastGE f r i)) 1 μ ≤ ↑(ENNReal.toNNReal (2 * ↑↑μ Set.univ * ENNReal.ofReal (r + ↑R))) ** refine' (hf.stoppedValue_leastGE_snorm_le hr hf0 hbdd i).trans _ ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hr : 0 ≤ r hf0 : f 0 = 0 hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R i : ℕ ⊢ 2 * ↑↑μ Set.univ * ENNReal.ofReal (r + ↑R) ≤ ↑(ENNReal.toNNReal (2 * ↑↑μ Set.univ * ENNReal.ofReal (r + ↑R))) ** simp [ENNReal.coe_toNNReal (measure_ne_top μ _), ENNReal.coe_toNNReal] ** Qed
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MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto_aux ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Submartingale f ℱ μ hf0 : f 0 = 0 hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R ⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) ** filter_upwards [hf.exists_tendsto_of_abs_bddAbove_aux hf0 hbdd] with ω hω using
⟨hω, fun ⟨c, hc⟩ => hc.bddAbove_range⟩ ** Qed
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MeasureTheory.Martingale.bddAbove_range_iff_bddBelow_range ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R ⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω) ** have hbdd' : ∀ᵐ ω ∂μ, ∀ i, |(-f) (i + 1) ω - (-f) i ω| ≤ R := by
filter_upwards [hbdd] with ω hω i
erw [← abs_neg, neg_sub, sub_neg_eq_add, neg_add_eq_sub]
exact hω i ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R ⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω) ** have hup := hf.submartingale.bddAbove_iff_exists_tendsto hbdd ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) ⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω) ** have hdown := hf.neg.submartingale.bddAbove_iff_exists_tendsto hbdd' ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω) ** filter_upwards [hup, hdown] with ω hω₁ hω₂ ** case h Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω : Ω hω₁ : BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) this : (∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ⊢ BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω) ** rw [hω₁, this, ← hω₂] ** case h Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω : Ω hω₁ : BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) this : (∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ⊢ BddAbove (Set.range fun n => (-f) n ω) ↔ BddBelow (Set.range fun n => f n ω) ** constructor <;> rintro ⟨c, hc⟩ <;> refine' ⟨-c, fun ω hω => _⟩ ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R ⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R ** filter_upwards [hbdd] with ω hω i ** case h Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R ω : Ω hω : ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R i : ℕ ⊢ |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R ** erw [← abs_neg, neg_sub, sub_neg_eq_add, neg_add_eq_sub] ** case h Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R ω : Ω hω : ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R i : ℕ ⊢ |f (i + 1) ω - f i ω| ≤ ↑R ** exact hω i ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω : Ω hω₁ : BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ⊢ (∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ** constructor <;> rintro ⟨c, hc⟩ ** case mp.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω : Ω hω₁ : BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) c : ℝ hc : Tendsto (fun n => f n ω) atTop (𝓝 c) ⊢ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ** exact ⟨-c, hc.neg⟩ ** case mpr.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω : Ω hω₁ : BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) c : ℝ hc : Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ⊢ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) ** refine' ⟨-c, _⟩ ** case mpr.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω : Ω hω₁ : BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) c : ℝ hc : Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ⊢ Tendsto (fun n => f n ω) atTop (𝓝 (-c)) ** convert hc.neg ** case h.e'_3.h Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω : Ω hω₁ : BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) c : ℝ hc : Tendsto (fun n => (-f) n ω) atTop (𝓝 c) x✝ : ℕ ⊢ f x✝ ω = -(-f) x✝ ω ** simp only [neg_neg, Pi.neg_apply] ** case h.mp.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝¹ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω✝ : Ω hω₁ : BddAbove (Set.range fun n => f n ω✝) ↔ ∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω✝) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) this : (∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) c : ℝ hc : c ∈ upperBounds (Set.range fun n => (-f) n ω✝) ω : ℝ hω : ω ∈ Set.range fun n => f n ω✝ ⊢ -c ≤ ω ** rw [mem_upperBounds] at hc ** case h.mp.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝¹ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω✝ : Ω hω₁ : BddAbove (Set.range fun n => f n ω✝) ↔ ∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω✝) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) this : (∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) c : ℝ hc : ∀ (x : ℝ), (x ∈ Set.range fun n => (-f) n ω✝) → x ≤ c ω : ℝ hω : ω ∈ Set.range fun n => f n ω✝ ⊢ -c ≤ ω ** refine' neg_le.2 (hc _ _) ** case h.mp.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝¹ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω✝ : Ω hω₁ : BddAbove (Set.range fun n => f n ω✝) ↔ ∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω✝) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) this : (∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) c : ℝ hc : ∀ (x : ℝ), (x ∈ Set.range fun n => (-f) n ω✝) → x ≤ c ω : ℝ hω : ω ∈ Set.range fun n => f n ω✝ ⊢ -ω ∈ Set.range fun n => (-f) n ω✝ ** simpa only [Pi.neg_apply, Set.mem_range, neg_inj] ** case h.mpr.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝¹ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω✝ : Ω hω₁ : BddAbove (Set.range fun n => f n ω✝) ↔ ∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω✝) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) this : (∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) c : ℝ hc : c ∈ lowerBounds (Set.range fun n => f n ω✝) ω : ℝ hω : ω ∈ Set.range fun n => (-f) n ω✝ ⊢ ω ≤ -c ** rw [mem_lowerBounds] at hc ** case h.mpr.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝¹ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω✝ : Ω hω₁ : BddAbove (Set.range fun n => f n ω✝) ↔ ∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω✝) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) this : (∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) c : ℝ hc : ∀ (x : ℝ), (x ∈ Set.range fun n => f n ω✝) → c ≤ x ω : ℝ hω : ω ∈ Set.range fun n => (-f) n ω✝ ⊢ ω ≤ -c ** simp_rw [Set.mem_range, Pi.neg_apply, neg_eq_iff_eq_neg] at hω ** case h.mpr.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝¹ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω✝ : Ω hω₁ : BddAbove (Set.range fun n => f n ω✝) ↔ ∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω✝) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) this : (∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) c : ℝ hc : ∀ (x : ℝ), (x ∈ Set.range fun n => f n ω✝) → c ≤ x ω : ℝ hω : ∃ y, f y ω✝ = -ω ⊢ ω ≤ -c ** refine' le_neg.1 (hc _ _) ** case h.mpr.intro Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝¹ : Ω r : ℝ R : ℝ≥0 inst✝ : IsFiniteMeasure μ hf : Martingale f ℱ μ hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c) ω✝ : Ω hω₁ : BddAbove (Set.range fun n => f n ω✝) ↔ ∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c) hω₂ : BddAbove (Set.range fun n => (-f) n ω✝) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) this : (∃ c, Tendsto (fun n => f n ω✝) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n => (-f) n ω✝) atTop (𝓝 c) c : ℝ hc : ∀ (x : ℝ), (x ∈ Set.range fun n => f n ω✝) → c ≤ x ω : ℝ hω : ∃ y, f y ω✝ = -ω ⊢ -ω ∈ Set.range fun n => f n ω✝ ** simpa only [Set.mem_range] ** Qed
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MeasureTheory.BorelCantelli.process_zero ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω : Ω r : ℝ R : ℝ≥0 s : ℕ → Set Ω ⊢ process s 0 = 0 ** rw [process, Finset.range_zero, Finset.sum_empty] ** Qed
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MeasureTheory.BorelCantelli.process_difference_le ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 s✝ s : ℕ → Set Ω ω : Ω n : ℕ ⊢ |process s (n + 1) ω - process s n ω| ≤ ↑1 ** norm_cast ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 s✝ s : ℕ → Set Ω ω : Ω n : ℕ ⊢ |process s (n + 1) ω - process s n ω| ≤ 1 ** rw [process, process, Finset.sum_apply, Finset.sum_apply,
Finset.sum_range_succ_sub_sum, ← Real.norm_eq_abs, norm_indicator_eq_indicator_norm] ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 s✝ s : ℕ → Set Ω ω : Ω n : ℕ ⊢ Set.indicator (s (n + 1)) (fun a => ‖OfNat.ofNat 1 a‖) ω ≤ 1 ** refine' Set.indicator_le' (fun _ _ => _) (fun _ _ => zero_le_one) _ ** Ω : Type u_1 m0 : MeasurableSpace Ω μ : Measure Ω ℱ : Filtration ℕ m0 f : ℕ → Ω → ℝ ω✝ : Ω r : ℝ R : ℝ≥0 s✝ s : ℕ → Set Ω ω : Ω n : ℕ x✝¹ : Ω x✝ : x✝¹ ∈ s (n + 1) ⊢ ‖OfNat.ofNat 1 x✝¹‖ ≤ 1 ** rw [Pi.one_apply, norm_one] ** Qed
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ProbabilityTheory.kernel.bind_add ** α : Type u_1 β : Type u_2 γ : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ μ ν : Measure α κ : { x // x ∈ kernel α β } ⊢ Measure.bind (μ + ν) ↑κ = Measure.bind μ ↑κ + Measure.bind ν ↑κ ** ext1 s hs ** case h α : Type u_1 β : Type u_2 γ : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ μ ν : Measure α κ : { x // x ∈ kernel α β } s : Set β hs : MeasurableSet s ⊢ ↑↑(Measure.bind (μ + ν) ↑κ) s = ↑↑(Measure.bind μ ↑κ + Measure.bind ν ↑κ) s ** rw [Measure.bind_apply hs (kernel.measurable _), lintegral_add_measure, Measure.coe_add,
Pi.add_apply, Measure.bind_apply hs (kernel.measurable _),
Measure.bind_apply hs (kernel.measurable _)] ** Qed
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ProbabilityTheory.kernel.bind_smul ** α : Type u_1 β : Type u_2 γ : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ κ : { x // x ∈ kernel α β } μ : Measure α r : ℝ≥0∞ ⊢ Measure.bind (r • μ) ↑κ = r • Measure.bind μ ↑κ ** ext1 s hs ** case h α : Type u_1 β : Type u_2 γ : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ κ : { x // x ∈ kernel α β } μ : Measure α r : ℝ≥0∞ s : Set β hs : MeasurableSet s ⊢ ↑↑(Measure.bind (r • μ) ↑κ) s = ↑↑(r • Measure.bind μ ↑κ) s ** rw [Measure.bind_apply hs (kernel.measurable _), lintegral_smul_measure, Measure.coe_smul,
Pi.smul_apply, Measure.bind_apply hs (kernel.measurable _), smul_eq_mul] ** Qed
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ProbabilityTheory.kernel.const_bind_eq_comp_const ** α : Type u_1 β : Type u_2 γ : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ κ : { x // x ∈ kernel α β } μ : Measure α ⊢ const α (Measure.bind μ ↑κ) = κ ∘ₖ const α μ ** ext a s hs ** case h.h α : Type u_1 β : Type u_2 γ : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ κ : { x // x ∈ kernel α β } μ : Measure α a : α s : Set β hs : MeasurableSet s ⊢ ↑↑(↑(const α (Measure.bind μ ↑κ)) a) s = ↑↑(↑(κ ∘ₖ const α μ) a) s ** simp_rw [comp_apply' _ _ _ hs, const_apply, Measure.bind_apply hs (kernel.measurable _)] ** Qed
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ProbabilityTheory.kernel.comp_const_apply_eq_bind ** α : Type u_1 β : Type u_2 γ : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ κ : { x // x ∈ kernel α β } μ : Measure α a : α ⊢ ↑(κ ∘ₖ const α μ) a = Measure.bind μ ↑κ ** rw [← const_apply (μ.bind κ) a, const_bind_eq_comp_const κ μ] ** Qed
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ProbabilityTheory.kernel.Invariant.comp_const ** α : Type u_1 β : Type u_2 γ : Type u_3 mα : MeasurableSpace α mβ : MeasurableSpace β mγ : MeasurableSpace γ κ η : { x // x ∈ kernel α α } μ : Measure α hκ : Invariant κ μ ⊢ κ ∘ₖ const α μ = const α μ ** rw [← const_bind_eq_comp_const κ μ, hκ.def] ** Qed
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MeasureTheory.martingale_const_fun ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ✝ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f✝ g : ι → Ω → E ℱ✝ : Filtration ι m0 inst✝¹ : OrderBot ι ℱ : Filtration ι m0 μ : Measure Ω inst✝ : IsFiniteMeasure μ f : Ω → E hf : StronglyMeasurable f hfint : Integrable f ⊢ Martingale (fun x => f) ℱ μ ** refine' ⟨fun i => hf.mono <| ℱ.mono bot_le, fun i j _ => _⟩ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ✝ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f✝ g : ι → Ω → E ℱ✝ : Filtration ι m0 inst✝¹ : OrderBot ι ℱ : Filtration ι m0 μ : Measure Ω inst✝ : IsFiniteMeasure μ f : Ω → E hf : StronglyMeasurable f hfint : Integrable f i j : ι x✝ : i ≤ j ⊢ μ[(fun x => f) j|↑ℱ i] =ᵐ[μ] (fun x => f) i ** rw [condexp_of_stronglyMeasurable (ℱ.le _) (hf.mono <| ℱ.mono bot_le) hfint] ** Qed
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MeasureTheory.Martingale.set_integral_eq ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : SigmaFiniteFiltration μ ℱ hf : Martingale f ℱ μ i j : ι hij : i ≤ j s : Set Ω hs : MeasurableSet s ⊢ ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (ω : Ω) in s, f j ω ∂μ ** rw [← @set_integral_condexp _ _ _ _ _ (ℱ i) m0 _ _ _ (ℱ.le i) _ (hf.integrable j) hs] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : SigmaFiniteFiltration μ ℱ hf : Martingale f ℱ μ i j : ι hij : i ≤ j s : Set Ω hs : MeasurableSet s ⊢ ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (x : Ω) in s, (μ[f j|↑ℱ i]) x ∂μ ** refine' set_integral_congr_ae (ℱ.le i s hs) _ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : SigmaFiniteFiltration μ ℱ hf : Martingale f ℱ μ i j : ι hij : i ≤ j s : Set Ω hs : MeasurableSet s ⊢ ∀ᵐ (x : Ω) ∂μ, x ∈ s → f i x = (μ[f j|↑ℱ i]) x ** filter_upwards [hf.2 i j hij] with _ heq _ using heq.symm ** Qed
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MeasureTheory.Martingale.add ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 hf : Martingale f ℱ μ hg : Martingale g ℱ μ ⊢ Martingale (f + g) ℱ μ ** refine' ⟨hf.adapted.add hg.adapted, fun i j hij => _⟩ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 hf : Martingale f ℱ μ hg : Martingale g ℱ μ i j : ι hij : i ≤ j ⊢ μ[(f + g) j|↑ℱ i] =ᵐ[μ] (f + g) i ** exact (condexp_add (hf.integrable j) (hg.integrable j)).trans ((hf.2 i j hij).add (hg.2 i j hij)) ** Qed
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MeasureTheory.Martingale.sub ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 hf : Martingale f ℱ μ hg : Martingale g ℱ μ ⊢ Martingale (f - g) ℱ μ ** rw [sub_eq_add_neg] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 hf : Martingale f ℱ μ hg : Martingale g ℱ μ ⊢ Martingale (f + -g) ℱ μ ** exact hf.add hg.neg ** Qed
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MeasureTheory.Martingale.smul ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 c : ℝ hf : Martingale f ℱ μ ⊢ Martingale (c • f) ℱ μ ** refine' ⟨hf.adapted.smul c, fun i j hij => _⟩ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 c : ℝ hf : Martingale f ℱ μ i j : ι hij : i ≤ j ⊢ μ[(c • f) j|↑ℱ i] =ᵐ[μ] (c • f) i ** refine' (condexp_smul c (f j)).trans ((hf.2 i j hij).mono fun x hx => _) ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 c : ℝ hf : Martingale f ℱ μ i j : ι hij : i ≤ j x : Ω hx : (μ[f j|↑ℱ i]) x = f i x ⊢ (c • μ[f j|↑ℱ i]) x = (c • f) i x ** rw [Pi.smul_apply, hx, Pi.smul_apply, Pi.smul_apply] ** Qed
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MeasureTheory.Supermartingale.set_integral_le ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : SigmaFiniteFiltration μ ℱ f : ι → Ω → ℝ hf : Supermartingale f ℱ μ i j : ι hij : i ≤ j s : Set Ω hs : MeasurableSet s ⊢ ∫ (ω : Ω) in s, f j ω ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ ** rw [← set_integral_condexp (ℱ.le i) (hf.integrable j) hs] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : SigmaFiniteFiltration μ ℱ f : ι → Ω → ℝ hf : Supermartingale f ℱ μ i j : ι hij : i ≤ j s : Set Ω hs : MeasurableSet s ⊢ ∫ (x : Ω) in s, (μ[f j|↑ℱ i]) x ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ ** refine' set_integral_mono_ae integrable_condexp.integrableOn (hf.integrable i).integrableOn _ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : SigmaFiniteFiltration μ ℱ f : ι → Ω → ℝ hf : Supermartingale f ℱ μ i j : ι hij : i ≤ j s : Set Ω hs : MeasurableSet s ⊢ (fun x => (μ[f j|↑ℱ i]) x) ≤ᵐ[μ] fun ω => f i ω ** filter_upwards [hf.2.1 i j hij] with _ heq using heq ** Qed
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MeasureTheory.Supermartingale.add ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ hg : Supermartingale g ℱ μ ⊢ Supermartingale (f + g) ℱ μ ** refine' ⟨hf.1.add hg.1, fun i j hij => _, fun i => (hf.2.2 i).add (hg.2.2 i)⟩ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ hg : Supermartingale g ℱ μ i j : ι hij : i ≤ j ⊢ μ[(f + g) j|↑ℱ i] ≤ᵐ[μ] (f + g) i ** refine' (condexp_add (hf.integrable j) (hg.integrable j)).le.trans _ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ hg : Supermartingale g ℱ μ i j : ι hij : i ≤ j ⊢ μ[f j|↑ℱ i] + μ[g j|↑ℱ i] ≤ᵐ[μ] (f + g) i ** filter_upwards [hf.2.1 i j hij, hg.2.1 i j hij] ** case h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ hg : Supermartingale g ℱ μ i j : ι hij : i ≤ j ⊢ ∀ (a : Ω), (μ[f j|↑ℱ i]) a ≤ f i a → (μ[g j|↑ℱ i]) a ≤ g i a → (μ[f j|↑ℱ i] + μ[g j|↑ℱ i]) a ≤ (f + g) i a ** intros ** case h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ hg : Supermartingale g ℱ μ i j : ι hij : i ≤ j a✝² : Ω a✝¹ : (μ[f j|↑ℱ i]) a✝² ≤ f i a✝² a✝ : (μ[g j|↑ℱ i]) a✝² ≤ g i a✝² ⊢ (μ[f j|↑ℱ i] + μ[g j|↑ℱ i]) a✝² ≤ (f + g) i a✝² ** refine' add_le_add _ _ <;> assumption ** Qed
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MeasureTheory.Supermartingale.neg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ ⊢ Submartingale (-f) ℱ μ ** refine' ⟨hf.1.neg, fun i j hij => _, fun i => (hf.2.2 i).neg⟩ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ i j : ι hij : i ≤ j ⊢ (-f) i ≤ᵐ[μ] μ[(-f) j|↑ℱ i] ** refine' EventuallyLE.trans _ (condexp_neg (f j)).symm.le ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ i j : ι hij : i ≤ j ⊢ (-f) i ≤ᵐ[μ] -μ[f j|↑ℱ i] ** filter_upwards [hf.2.1 i j hij] with _ _ ** case h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ i j : ι hij : i ≤ j a✝¹ : Ω a✝ : (μ[f j|↑ℱ i]) a✝¹ ≤ f i a✝¹ ⊢ (-f) i a✝¹ ≤ (-μ[f j|↑ℱ i]) a✝¹ ** simpa ** Qed
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MeasureTheory.Submartingale.add ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ hg : Submartingale g ℱ μ ⊢ Submartingale (f + g) ℱ μ ** refine' ⟨hf.1.add hg.1, fun i j hij => _, fun i => (hf.2.2 i).add (hg.2.2 i)⟩ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ hg : Submartingale g ℱ μ i j : ι hij : i ≤ j ⊢ (f + g) i ≤ᵐ[μ] μ[(f + g) j|↑ℱ i] ** refine' EventuallyLE.trans _ (condexp_add (hf.integrable j) (hg.integrable j)).symm.le ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ hg : Submartingale g ℱ μ i j : ι hij : i ≤ j ⊢ (f + g) i ≤ᵐ[μ] μ[f j|↑ℱ i] + μ[g j|↑ℱ i] ** filter_upwards [hf.2.1 i j hij, hg.2.1 i j hij] ** case h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ hg : Submartingale g ℱ μ i j : ι hij : i ≤ j ⊢ ∀ (a : Ω), f i a ≤ (μ[f j|↑ℱ i]) a → g i a ≤ (μ[g j|↑ℱ i]) a → (f + g) i a ≤ (μ[f j|↑ℱ i] + μ[g j|↑ℱ i]) a ** intros ** case h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ hg : Submartingale g ℱ μ i j : ι hij : i ≤ j a✝² : Ω a✝¹ : f i a✝² ≤ (μ[f j|↑ℱ i]) a✝² a✝ : g i a✝² ≤ (μ[g j|↑ℱ i]) a✝² ⊢ (f + g) i a✝² ≤ (μ[f j|↑ℱ i] + μ[g j|↑ℱ i]) a✝² ** refine' add_le_add _ _ <;> assumption ** Qed
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MeasureTheory.Submartingale.neg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ ⊢ Supermartingale (-f) ℱ μ ** refine' ⟨hf.1.neg, fun i j hij => (condexp_neg (f j)).le.trans _, fun i => (hf.2.2 i).neg⟩ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ i j : ι hij : i ≤ j ⊢ -μ[f j|↑ℱ i] ≤ᵐ[μ] (-f) i ** filter_upwards [hf.2.1 i j hij] with _ _ ** case h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ i j : ι hij : i ≤ j a✝¹ : Ω a✝ : f i a✝¹ ≤ (μ[f j|↑ℱ i]) a✝¹ ⊢ (-μ[f j|↑ℱ i]) a✝¹ ≤ (-f) i a✝¹ ** simpa ** Qed
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MeasureTheory.Submartingale.sub_supermartingale ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ hg : Supermartingale g ℱ μ ⊢ Submartingale (f - g) ℱ μ ** rw [sub_eq_add_neg] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Submartingale f ℱ μ hg : Supermartingale g ℱ μ ⊢ Submartingale (f + -g) ℱ μ ** exact hf.add hg.neg ** Qed
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MeasureTheory.submartingale_of_set_integral_le ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ ⊢ Submartingale f ℱ μ ** refine' ⟨hadp, fun i j hij => _, hint⟩ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ i j : ι hij : i ≤ j ⊢ f i ≤ᵐ[μ] μ[f j|↑ℱ i] ** suffices f i ≤ᵐ[μ.trim (ℱ.le i)] μ[f j|ℱ i] by exact ae_le_of_ae_le_trim this ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ i j : ι hij : i ≤ j ⊢ f i ≤ᵐ[Measure.trim μ (_ : ↑ℱ i ≤ m0)] μ[f j|↑ℱ i] ** suffices 0 ≤ᵐ[μ.trim (ℱ.le i)] μ[f j|ℱ i] - f i by
filter_upwards [this] with x hx
rwa [← sub_nonneg] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ i j : ι hij : i ≤ j ⊢ 0 ≤ᵐ[Measure.trim μ (_ : ↑ℱ i ≤ m0)] μ[f j|↑ℱ i] - f i ** refine' ae_nonneg_of_forall_set_integral_nonneg
((integrable_condexp.sub (hint i)).trim _ (stronglyMeasurable_condexp.sub <| hadp i))
fun s hs _ => _ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ i j : ι hij : i ≤ j s : Set Ω hs : MeasurableSet s x✝ : ↑↑(Measure.trim μ (_ : ↑ℱ i ≤ m0)) s < ⊤ ⊢ 0 ≤ ∫ (x : Ω) in s, (μ[f j|↑ℱ i] - f i) x ∂Measure.trim μ (_ : ↑ℱ i ≤ m0) ** specialize hf i j hij s hs ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) i j : ι hij : i ≤ j s : Set Ω hs : MeasurableSet s x✝ : ↑↑(Measure.trim μ (_ : ↑ℱ i ≤ m0)) s < ⊤ hf : ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ ⊢ 0 ≤ ∫ (x : Ω) in s, (μ[f j|↑ℱ i] - f i) x ∂Measure.trim μ (_ : ↑ℱ i ≤ m0) ** rwa [← set_integral_trim _ (stronglyMeasurable_condexp.sub <| hadp i) hs,
integral_sub' integrable_condexp.integrableOn (hint i).integrableOn, sub_nonneg,
set_integral_condexp (ℱ.le i) (hint j) hs] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ i j : ι hij : i ≤ j this : f i ≤ᵐ[Measure.trim μ (_ : ↑ℱ i ≤ m0)] μ[f j|↑ℱ i] ⊢ f i ≤ᵐ[μ] μ[f j|↑ℱ i] ** exact ae_le_of_ae_le_trim this ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ i j : ι hij : i ≤ j this : 0 ≤ᵐ[Measure.trim μ (_ : ↑ℱ i ≤ m0)] μ[f j|↑ℱ i] - f i ⊢ f i ≤ᵐ[Measure.trim μ (_ : ↑ℱ i ≤ m0)] μ[f j|↑ℱ i] ** filter_upwards [this] with x hx ** case h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ i j : ι hij : i ≤ j this : 0 ≤ᵐ[Measure.trim μ (_ : ↑ℱ i ≤ m0)] μ[f j|↑ℱ i] - f i x : Ω hx : OfNat.ofNat 0 x ≤ (μ[f j|↑ℱ i] - f i) x ⊢ f i x ≤ (μ[f j|↑ℱ i]) x ** rwa [← sub_nonneg] ** Qed
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MeasureTheory.submartingale_of_condexp_sub_nonneg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i] ⊢ Submartingale f ℱ μ ** refine' ⟨hadp, fun i j hij => _, hint⟩ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i] i j : ι hij : i ≤ j ⊢ f i ≤ᵐ[μ] μ[f j|↑ℱ i] ** rw [← condexp_of_stronglyMeasurable (ℱ.le _) (hadp _) (hint _), ← eventually_sub_nonneg] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 inst✝ : IsFiniteMeasure μ f : ι → Ω → ℝ hadp : Adapted ℱ f hint : ∀ (i : ι), Integrable (f i) hf : ∀ (i j : ι), i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i] i j : ι hij : i ≤ j ⊢ 0 ≤ᵐ[μ] μ[f j|↑ℱ i] - μ[f i|↑ℱ i] ** exact EventuallyLE.trans (hf i j hij) (condexp_sub (hint _) (hint _)).le ** Qed
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MeasureTheory.Submartingale.condexp_sub_nonneg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 f : ι → Ω → ℝ hf : Submartingale f ℱ μ i j : ι hij : i ≤ j ⊢ 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i] ** by_cases h : SigmaFinite (μ.trim (ℱ.le i)) ** case pos Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 f : ι → Ω → ℝ hf : Submartingale f ℱ μ i j : ι hij : i ≤ j h : SigmaFinite (Measure.trim μ (_ : ↑ℱ i ≤ m0)) ⊢ 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i] case neg Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 f : ι → Ω → ℝ hf : Submartingale f ℱ μ i j : ι hij : i ≤ j h : ¬SigmaFinite (Measure.trim μ (_ : ↑ℱ i ≤ m0)) ⊢ 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i] ** swap ** case pos Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 f : ι → Ω → ℝ hf : Submartingale f ℱ μ i j : ι hij : i ≤ j h : SigmaFinite (Measure.trim μ (_ : ↑ℱ i ≤ m0)) ⊢ 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i] ** refine' EventuallyLE.trans _ (condexp_sub (hf.integrable _) (hf.integrable _)).symm.le ** case pos Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 f : ι → Ω → ℝ hf : Submartingale f ℱ μ i j : ι hij : i ≤ j h : SigmaFinite (Measure.trim μ (_ : ↑ℱ i ≤ m0)) ⊢ 0 ≤ᵐ[μ] μ[f j|↑ℱ i] - μ[f i|↑ℱ i] ** rw [eventually_sub_nonneg,
condexp_of_stronglyMeasurable (ℱ.le _) (hf.adapted _) (hf.integrable _)] ** case neg Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 f : ι → Ω → ℝ hf : Submartingale f ℱ μ i j : ι hij : i ≤ j h : ¬SigmaFinite (Measure.trim μ (_ : ↑ℱ i ≤ m0)) ⊢ 0 ≤ᵐ[μ] μ[f j - f i|↑ℱ i] ** rw [condexp_of_not_sigmaFinite (ℱ.le i) h] ** case pos Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝³ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 f : ι → Ω → ℝ hf : Submartingale f ℱ μ i j : ι hij : i ≤ j h : SigmaFinite (Measure.trim μ (_ : ↑ℱ i ≤ m0)) ⊢ f i ≤ᵐ[μ] μ[f j|↑ℱ i] ** exact hf.2.1 i j hij ** Qed
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MeasureTheory.Supermartingale.sub_submartingale ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ hg : Submartingale g ℱ μ ⊢ Supermartingale (f - g) ℱ μ ** rw [sub_eq_add_neg] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f g : ι → Ω → E ℱ : Filtration ι m0 inst✝¹ : Preorder E inst✝ : CovariantClass E E (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 hf : Supermartingale f ℱ μ hg : Submartingale g ℱ μ ⊢ Supermartingale (f + -g) ℱ μ ** exact hf.add hg.neg ** Qed
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MeasureTheory.Supermartingale.smul_nonpos ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : c ≤ 0 hf : Supermartingale f ℱ μ ⊢ Submartingale (c • f) ℱ μ ** rw [← neg_neg c, (by ext (i x); simp : - -c • f = -(-c • f))] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : c ≤ 0 hf : Supermartingale f ℱ μ ⊢ Submartingale (-(-c • f)) ℱ μ ** exact (hf.smul_nonneg <| neg_nonneg.2 hc).neg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : c ≤ 0 hf : Supermartingale f ℱ μ ⊢ - -c • f = -(-c • f) ** ext (i x) ** case h.h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : c ≤ 0 hf : Supermartingale f ℱ μ i : ι x : Ω ⊢ (- -c • f) i x = (-(-c • f)) i x ** simp ** Qed
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MeasureTheory.Submartingale.smul_nonneg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : 0 ≤ c hf : Submartingale f ℱ μ ⊢ Submartingale (c • f) ℱ μ ** rw [← neg_neg c, (by ext (i x); simp : - -c • f = -(c • -f))] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : 0 ≤ c hf : Submartingale f ℱ μ ⊢ Submartingale (-(c • -f)) ℱ μ ** exact Supermartingale.neg (hf.neg.smul_nonneg hc) ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : 0 ≤ c hf : Submartingale f ℱ μ ⊢ - -c • f = -(c • -f) ** ext (i x) ** case h.h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : 0 ≤ c hf : Submartingale f ℱ μ i : ι x : Ω ⊢ (- -c • f) i x = (-(c • -f)) i x ** simp ** Qed
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MeasureTheory.Submartingale.smul_nonpos ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : c ≤ 0 hf : Submartingale f ℱ μ ⊢ Supermartingale (c • f) ℱ μ ** rw [← neg_neg c, (by ext (i x); simp : - -c • f = -(-c • f))] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : c ≤ 0 hf : Submartingale f ℱ μ ⊢ Supermartingale (-(-c • f)) ℱ μ ** exact (hf.smul_nonneg <| neg_nonneg.2 hc).neg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : c ≤ 0 hf : Submartingale f ℱ μ ⊢ - -c • f = -(-c • f) ** ext (i x) ** case h.h Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁷ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁶ : NormedAddCommGroup E inst✝⁵ : NormedSpace ℝ E inst✝⁴ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 F : Type u_4 inst✝³ : NormedLatticeAddCommGroup F inst✝² : NormedSpace ℝ F inst✝¹ : CompleteSpace F inst✝ : OrderedSMul ℝ F f : ι → Ω → F c : ℝ hc : c ≤ 0 hf : Submartingale f ℱ μ i : ι x : Ω ⊢ (- -c • f) i x = (-(-c • f)) i x ** simp ** Qed
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MeasureTheory.submartingale_of_set_integral_le_succ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ ⊢ Submartingale f 𝒢 μ ** refine' submartingale_of_set_integral_le hadp hint fun i j hij s hs => _ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ i j : ℕ hij : i ≤ j s : Set Ω hs : MeasurableSet s ⊢ ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ ** induction' hij with k hk₁ hk₂ ** case refl Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ i j : ℕ s : Set Ω hs : MeasurableSet s ⊢ ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ ** exact le_rfl ** case step Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ i j : ℕ s : Set Ω hs : MeasurableSet s k : ℕ hk₁ : Nat.le i k hk₂ : ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f k ω ∂μ ⊢ ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (Nat.succ k) ω ∂μ ** exact le_trans hk₂ (hf k s (𝒢.mono hk₁ _ hs)) ** Qed
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MeasureTheory.supermartingale_of_set_integral_succ_le ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f (i + 1) ω ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ ⊢ Supermartingale f 𝒢 μ ** rw [← neg_neg f] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f (i + 1) ω ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ ⊢ Supermartingale (- -f) 𝒢 μ ** refine' (submartingale_of_set_integral_le_succ hadp.neg (fun i => (hint i).neg) _).neg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f (i + 1) ω ∂μ ≤ ∫ (ω : Ω) in s, f i ω ∂μ ⊢ ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, (-f) i ω ∂μ ≤ ∫ (ω : Ω) in s, (-f) (i + 1) ω ∂μ ** simpa only [integral_neg, Pi.neg_apply, neg_le_neg_iff] ** Qed
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MeasureTheory.submartingale_nat ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), f i ≤ᵐ[μ] μ[f (i + 1)|↑𝒢 i] ⊢ Submartingale f 𝒢 μ ** refine' submartingale_of_set_integral_le_succ hadp hint fun i s hs => _ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), f i ≤ᵐ[μ] μ[f (i + 1)|↑𝒢 i] i : ℕ s : Set Ω hs : MeasurableSet s ⊢ ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ ** have : ∫ ω in s, f (i + 1) ω ∂μ = ∫ ω in s, (μ[f (i + 1)|𝒢 i]) ω ∂μ :=
(set_integral_condexp (𝒢.le i) (hint _) hs).symm ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), f i ≤ᵐ[μ] μ[f (i + 1)|↑𝒢 i] i : ℕ s : Set Ω hs : MeasurableSet s this : ∫ (ω : Ω) in s, f (i + 1) ω ∂μ = ∫ (ω : Ω) in s, (μ[f (i + 1)|↑𝒢 i]) ω ∂μ ⊢ ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ ** rw [this] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), f i ≤ᵐ[μ] μ[f (i + 1)|↑𝒢 i] i : ℕ s : Set Ω hs : MeasurableSet s this : ∫ (ω : Ω) in s, f (i + 1) ω ∂μ = ∫ (ω : Ω) in s, (μ[f (i + 1)|↑𝒢 i]) ω ∂μ ⊢ ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, (μ[f (i + 1)|↑𝒢 i]) ω ∂μ ** exact set_integral_mono_ae (hint i).integrableOn integrable_condexp.integrableOn (hf i) ** Qed
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MeasureTheory.supermartingale_nat ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), μ[f (i + 1)|↑𝒢 i] ≤ᵐ[μ] f i ⊢ Supermartingale f 𝒢 μ ** rw [← neg_neg f] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), μ[f (i + 1)|↑𝒢 i] ≤ᵐ[μ] f i ⊢ Supermartingale (- -f) 𝒢 μ ** refine' (submartingale_nat hadp.neg (fun i => (hint i).neg) fun i =>
EventuallyLE.trans _ (condexp_neg _).symm.le).neg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), μ[f (i + 1)|↑𝒢 i] ≤ᵐ[μ] f i i : ℕ ⊢ (-f) i ≤ᵐ[μ] -μ[fun i_1 => f (i + 1) i_1|↑𝒢 i] ** filter_upwards [hf i] with x hx using neg_le_neg hx ** Qed
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MeasureTheory.submartingale_of_condexp_sub_nonneg_nat ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), 0 ≤ᵐ[μ] μ[f (i + 1) - f i|↑𝒢 i] ⊢ Submartingale f 𝒢 μ ** refine' submartingale_nat hadp hint fun i => _ ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), 0 ≤ᵐ[μ] μ[f (i + 1) - f i|↑𝒢 i] i : ℕ ⊢ f i ≤ᵐ[μ] μ[f (i + 1)|↑𝒢 i] ** rw [← condexp_of_stronglyMeasurable (𝒢.le _) (hadp _) (hint _), ← eventually_sub_nonneg] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), 0 ≤ᵐ[μ] μ[f (i + 1) - f i|↑𝒢 i] i : ℕ ⊢ 0 ≤ᵐ[μ] μ[f (i + 1)|↑𝒢 i] - μ[f i|↑𝒢 i] ** exact EventuallyLE.trans (hf i) (condexp_sub (hint _) (hint _)).le ** Qed
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MeasureTheory.supermartingale_of_condexp_sub_nonneg_nat ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), 0 ≤ᵐ[μ] μ[f i - f (i + 1)|↑𝒢 i] ⊢ Supermartingale f 𝒢 μ ** rw [← neg_neg f] ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), 0 ≤ᵐ[μ] μ[f i - f (i + 1)|↑𝒢 i] ⊢ Supermartingale (- -f) 𝒢 μ ** refine' (submartingale_of_condexp_sub_nonneg_nat hadp.neg (fun i => (hint i).neg) _).neg ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : IsFiniteMeasure μ f : ℕ → Ω → ℝ hadp : Adapted 𝒢 f hint : ∀ (i : ℕ), Integrable (f i) hf : ∀ (i : ℕ), 0 ≤ᵐ[μ] μ[f i - f (i + 1)|↑𝒢 i] ⊢ ∀ (i : ℕ), 0 ≤ᵐ[μ] μ[(-f) (i + 1) - (-f) i|↑𝒢 i] ** simpa only [Pi.zero_apply, Pi.neg_apply, neg_sub_neg] ** Qed
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MeasureTheory.Submartingale.zero_le_of_predictable ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝¹ : Preorder E inst✝ : SigmaFiniteFiltration μ 𝒢 f : ℕ → Ω → E hfmgle : Submartingale f 𝒢 μ hfadp : Adapted 𝒢 fun n => f (n + 1) n : ℕ ⊢ f 0 ≤ᵐ[μ] f n ** induction' n with k ih ** case zero Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝¹ : Preorder E inst✝ : SigmaFiniteFiltration μ 𝒢 f : ℕ → Ω → E hfmgle : Submartingale f 𝒢 μ hfadp : Adapted 𝒢 fun n => f (n + 1) ⊢ f 0 ≤ᵐ[μ] f Nat.zero ** rfl ** case succ Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝¹ : Preorder E inst✝ : SigmaFiniteFiltration μ 𝒢 f : ℕ → Ω → E hfmgle : Submartingale f 𝒢 μ hfadp : Adapted 𝒢 fun n => f (n + 1) k : ℕ ih : f 0 ≤ᵐ[μ] f k ⊢ f 0 ≤ᵐ[μ] f (Nat.succ k) ** exact ih.trans ((hfmgle.2.1 k (k + 1) k.le_succ).trans_eq <| Germ.coe_eq.mp <|
congr_arg Germ.ofFun <| condexp_of_stronglyMeasurable (𝒢.le _) (hfadp _) <| hfmgle.integrable _) ** Qed
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MeasureTheory.Supermartingale.le_zero_of_predictable ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝¹ : Preorder E inst✝ : SigmaFiniteFiltration μ 𝒢 f : ℕ → Ω → E hfmgle : Supermartingale f 𝒢 μ hfadp : Adapted 𝒢 fun n => f (n + 1) n : ℕ ⊢ f n ≤ᵐ[μ] f 0 ** induction' n with k ih ** case zero Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝¹ : Preorder E inst✝ : SigmaFiniteFiltration μ 𝒢 f : ℕ → Ω → E hfmgle : Supermartingale f 𝒢 μ hfadp : Adapted 𝒢 fun n => f (n + 1) ⊢ f Nat.zero ≤ᵐ[μ] f 0 ** rfl ** case succ Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁵ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝¹ : Preorder E inst✝ : SigmaFiniteFiltration μ 𝒢 f : ℕ → Ω → E hfmgle : Supermartingale f 𝒢 μ hfadp : Adapted 𝒢 fun n => f (n + 1) k : ℕ ih : f k ≤ᵐ[μ] f 0 ⊢ f (Nat.succ k) ≤ᵐ[μ] f 0 ** exact ((Germ.coe_eq.mp <| congr_arg Germ.ofFun <| condexp_of_stronglyMeasurable (𝒢.le _)
(hfadp _) <| hfmgle.integrable _).symm.trans_le (hfmgle.2.1 k (k + 1) k.le_succ)).trans ih ** Qed
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MeasureTheory.Martingale.eq_zero_of_predictable ** Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : SigmaFiniteFiltration μ 𝒢 f : ℕ → Ω → E hfmgle : Martingale f 𝒢 μ hfadp : Adapted 𝒢 fun n => f (n + 1) n : ℕ ⊢ f n =ᵐ[μ] f 0 ** induction' n with k ih ** case zero Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : SigmaFiniteFiltration μ 𝒢 f : ℕ → Ω → E hfmgle : Martingale f 𝒢 μ hfadp : Adapted 𝒢 fun n => f (n + 1) ⊢ f Nat.zero =ᵐ[μ] f 0 ** rfl ** case succ Ω : Type u_1 E : Type u_2 ι : Type u_3 inst✝⁴ : Preorder ι m0 : MeasurableSpace Ω μ : Measure Ω inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E f✝ g : ι → Ω → E ℱ : Filtration ι m0 𝒢 : Filtration ℕ m0 inst✝ : SigmaFiniteFiltration μ 𝒢 f : ℕ → Ω → E hfmgle : Martingale f 𝒢 μ hfadp : Adapted 𝒢 fun n => f (n + 1) k : ℕ ih : f k =ᵐ[μ] f 0 ⊢ f (Nat.succ k) =ᵐ[μ] f 0 ** exact ((Germ.coe_eq.mp (congr_arg Germ.ofFun <| condexp_of_stronglyMeasurable (𝒢.le _) (hfadp _)
(hfmgle.integrable _))).symm.trans (hfmgle.2 k (k + 1) k.le_succ)).trans ih ** Qed
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MeasureTheory.predictablePart_zero ** Ω : Type u_1 E : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f : ℕ → Ω → E ℱ : Filtration ℕ m0 n : ℕ ⊢ predictablePart f ℱ μ 0 = 0 ** simp_rw [predictablePart, Finset.range_zero, Finset.sum_empty] ** Qed
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MeasureTheory.integrable_martingalePart ** Ω : Type u_1 E : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f : ℕ → Ω → E ℱ : Filtration ℕ m0 n✝ : ℕ hf_int : ∀ (n : ℕ), Integrable (f n) n : ℕ ⊢ Integrable (martingalePart f ℱ μ n) ** rw [martingalePart_eq_sum] ** Ω : Type u_1 E : Type u_2 m0 : MeasurableSpace Ω μ : Measure Ω inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f : ℕ → Ω → E ℱ : Filtration ℕ m0 n✝ : ℕ hf_int : ∀ (n : ℕ), Integrable (f n) n : ℕ ⊢ Integrable ((fun n => f 0 + ∑ i in Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])) n) ** exact (hf_int 0).add
(integrable_finset_sum' _ fun i _ => ((hf_int _).sub (hf_int _)).sub integrable_condexp) ** Qed
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ProbabilityTheory.set_lintegral_condKernelReal_univ ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α) in s, ↑↑(↑(condKernelReal ρ) a) univ ∂Measure.fst ρ = ↑↑ρ (s ×ˢ univ) ** simp only [measure_univ, lintegral_const, Measure.restrict_apply, MeasurableSet.univ, univ_inter,
one_mul, Measure.fst_apply hs, ← prod_univ] ** Qed
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ProbabilityTheory.lintegral_condKernelReal_univ ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) ⊢ ∫⁻ (a : α), ↑↑(↑(condKernelReal ρ) a) univ ∂Measure.fst ρ = ↑↑ρ univ ** rw [← set_lintegral_univ, set_lintegral_condKernelReal_univ ρ MeasurableSet.univ,
univ_prod_univ] ** Qed
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ProbabilityTheory.measure_eq_compProd_real ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ ⊢ ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (condKernelReal ρ)) () ** rw [← kernel.const_eq_compProd_real Unit ρ, kernel.const_apply] ** Qed
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ProbabilityTheory.lintegral_condKernelReal ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ f : α × ℝ → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), ∫⁻ (y : ℝ), f (a, y) ∂↑(condKernelReal ρ) a ∂Measure.fst ρ = ∫⁻ (x : α × ℝ), f x ∂ρ ** nth_rw 3 [measure_eq_compProd_real ρ] ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ f : α × ℝ → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), ∫⁻ (y : ℝ), f (a, y) ∂↑(condKernelReal ρ) a ∂Measure.fst ρ = ∫⁻ (x : α × ℝ), f x ∂↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (condKernelReal ρ)) () ** rw [kernel.lintegral_compProd _ _ _ hf, kernel.const_apply] ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ f : α × ℝ → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), ∫⁻ (y : ℝ), f (a, y) ∂↑(condKernelReal ρ) a ∂Measure.fst ρ = ∫⁻ (b : α), ∫⁻ (c : ℝ), f (b, c) ∂↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b) ∂Measure.fst ρ ** simp_rw [kernel.prodMkLeft_apply] ** Qed
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ProbabilityTheory.ae_condKernelReal_eq_one ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, ↑↑(↑(condKernelReal ρ) a) s = 1 ** have h : ρ {x | x.snd ∈ sᶜ} = (kernel.const Unit ρ.fst ⊗ₖ
kernel.prodMkLeft Unit (condKernelReal ρ)) () {x | x.snd ∈ sᶜ} := by
rw [← measure_eq_compProd_real] ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : ↑↑ρ {x | x.2 ∈ sᶜ} = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (condKernelReal ρ)) ()) {x | x.2 ∈ sᶜ} ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, ↑↑(↑(condKernelReal ρ) a) s = 1 ** rw [hρ, kernel.compProd_apply] at h ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : 0 = ∫⁻ (b : α), ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}} ∂↑(kernel.const Unit (Measure.fst ρ)) () ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, ↑↑(↑(condKernelReal ρ) a) s = 1 case hs α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : 0 = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (condKernelReal ρ)) ()) {x | x.2 ∈ sᶜ} ⊢ MeasurableSet {x | x.2 ∈ sᶜ} ** swap ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : 0 = ∫⁻ (b : α), ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}} ∂↑(kernel.const Unit (Measure.fst ρ)) () ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, ↑↑(↑(condKernelReal ρ) a) s = 1 ** rw [eq_comm, lintegral_eq_zero_iff] at h ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : (fun b => ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}}) =ᵐ[↑(kernel.const Unit (Measure.fst ρ)) ()] 0 ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, ↑↑(↑(condKernelReal ρ) a) s = 1 α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : ∫⁻ (b : α), ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}} ∂↑(kernel.const Unit (Measure.fst ρ)) () = 0 ⊢ Measurable fun b => ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}} ** swap ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : (fun b => ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}}) =ᵐ[↑(kernel.const Unit (Measure.fst ρ)) ()] 0 ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, ↑↑(↑(condKernelReal ρ) a) s = 1 ** rw [kernel.const_apply] at h ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : (fun b => ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}}) =ᵐ[Measure.fst ρ] 0 ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, ↑↑(↑(condKernelReal ρ) a) s = 1 ** simp only [mem_compl_iff, mem_setOf_eq, kernel.prodMkLeft_apply'] at h ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : (fun b => ↑↑(↑(condKernelReal ρ) b) {c | ¬c ∈ s}) =ᵐ[Measure.fst ρ] 0 ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, ↑↑(↑(condKernelReal ρ) a) s = 1 ** filter_upwards [h] with a ha ** case h α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : (fun b => ↑↑(↑(condKernelReal ρ) b) {c | ¬c ∈ s}) =ᵐ[Measure.fst ρ] 0 a : α ha : ↑↑(↑(condKernelReal ρ) a) {c | ¬c ∈ s} = OfNat.ofNat 0 a ⊢ ↑↑(↑(condKernelReal ρ) a) s = 1 ** change condKernelReal ρ a sᶜ = 0 at ha ** case h α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : (fun b => ↑↑(↑(condKernelReal ρ) b) {c | ¬c ∈ s}) =ᵐ[Measure.fst ρ] 0 a : α ha : ↑↑(↑(condKernelReal ρ) a) sᶜ = 0 ⊢ ↑↑(↑(condKernelReal ρ) a) s = 1 ** rwa [prob_compl_eq_zero_iff hs] at ha ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 ⊢ ↑↑ρ {x | x.2 ∈ sᶜ} = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (condKernelReal ρ)) ()) {x | x.2 ∈ sᶜ} ** rw [← measure_eq_compProd_real] ** case hs α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : 0 = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (condKernelReal ρ)) ()) {x | x.2 ∈ sᶜ} ⊢ MeasurableSet {x | x.2 ∈ sᶜ} ** exact measurable_snd hs.compl ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : ∫⁻ (b : α), ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}} ∂↑(kernel.const Unit (Measure.fst ρ)) () = 0 ⊢ Measurable fun b => ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}} ** simp_rw [kernel.prodMkLeft_apply'] ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : ∫⁻ (b : α), ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}} ∂↑(kernel.const Unit (Measure.fst ρ)) () = 0 ⊢ Measurable fun b => ↑↑(↑(condKernelReal ρ) b) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}} ** simp only [mem_compl_iff, mem_setOf_eq] ** α : Type u_1 mα : MeasurableSpace α ρ : Measure (α × ℝ) inst✝ : IsFiniteMeasure ρ s : Set ℝ hs : MeasurableSet s hρ : ↑↑ρ {x | x.2 ∈ sᶜ} = 0 h : ∫⁻ (b : α), ↑↑(↑(kernel.prodMkLeft Unit (condKernelReal ρ)) ((), b)) {c | (b, c) ∈ {x | x.2 ∈ sᶜ}} ∂↑(kernel.const Unit (Measure.fst ρ)) () = 0 ⊢ Measurable fun b => ↑↑(↑(condKernelReal ρ) b) {c | ¬c ∈ s} ** exact kernel.measurable_coe _ hs.compl ** Qed
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ProbabilityTheory.condKernel_def ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ ⊢ Measure.condKernel ρ = Exists.choose (_ : ∃ η _h, kernel.const Unit ρ = kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit η) ** rw [MeasureTheory.Measure.condKernel] ** Qed
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ProbabilityTheory.measure_eq_compProd ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ ⊢ ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) () ** rw [← kernel.const_unit_eq_compProd, kernel.const_apply] ** Qed
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ProbabilityTheory.kernel.const_eq_compProd ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁷ : TopologicalSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : MeasurableSpace Ω inst✝⁴ : BorelSpace Ω inst✝³ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝² : IsFiniteMeasure ρ✝ γ : Type u_3 inst✝¹ : MeasurableSpace γ ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ ⊢ const γ ρ = const γ (Measure.fst ρ) ⊗ₖ prodMkLeft γ (Measure.condKernel ρ) ** ext a s hs : 2 ** case h.h α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁷ : TopologicalSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : MeasurableSpace Ω inst✝⁴ : BorelSpace Ω inst✝³ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝² : IsFiniteMeasure ρ✝ γ : Type u_3 inst✝¹ : MeasurableSpace γ ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ a : γ s : Set (α × Ω) hs : MeasurableSet s ⊢ ↑↑(↑(const γ ρ) a) s = ↑↑(↑(const γ (Measure.fst ρ) ⊗ₖ prodMkLeft γ (Measure.condKernel ρ)) a) s ** simpa only [kernel.const_apply, kernel.compProd_apply _ _ _ hs, kernel.prodMkLeft_apply'] using
kernel.ext_iff'.mp (kernel.const_unit_eq_compProd ρ) () s hs ** Qed
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ProbabilityTheory.lintegral_condKernel_mem ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set (α × Ω) hs : MeasurableSet s ⊢ ∫⁻ (a : α), ↑↑(↑(Measure.condKernel ρ) a) {x | (a, x) ∈ s} ∂Measure.fst ρ = ↑↑ρ s ** conv_rhs => rw [measure_eq_compProd ρ] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set (α × Ω) hs : MeasurableSet s ⊢ ∫⁻ (a : α), ↑↑(↑(Measure.condKernel ρ) a) {x | (a, x) ∈ s} ∂Measure.fst ρ = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) s ** simp_rw [kernel.compProd_apply _ _ _ hs, kernel.const_apply, kernel.prodMkLeft_apply] ** Qed
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ProbabilityTheory.set_lintegral_condKernel_eq_measure_prod ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t ⊢ ∫⁻ (a : α) in s, ↑↑(↑(Measure.condKernel ρ) a) t ∂Measure.fst ρ = ↑↑ρ (s ×ˢ t) ** have : ρ (s ×ˢ t) =
((kernel.const Unit ρ.fst ⊗ₖ kernel.prodMkLeft Unit ρ.condKernel) ()) (s ×ˢ t) := by
congr; exact measure_eq_compProd ρ ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t this : ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) ⊢ ∫⁻ (a : α) in s, ↑↑(↑(Measure.condKernel ρ) a) t ∂Measure.fst ρ = ↑↑ρ (s ×ˢ t) ** rw [this, kernel.compProd_apply _ _ _ (hs.prod ht)] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t this : ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) ⊢ ∫⁻ (a : α) in s, ↑↑(↑(Measure.condKernel ρ) a) t ∂Measure.fst ρ = ∫⁻ (b : α), ↑↑(↑(kernel.prodMkLeft Unit (Measure.condKernel ρ)) ((), b)) {c | (b, c) ∈ s ×ˢ t} ∂↑(kernel.const Unit (Measure.fst ρ)) () ** simp only [prod_mk_mem_set_prod_eq, kernel.lintegral_const, kernel.prodMkLeft_apply] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t this : ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) ⊢ ∫⁻ (a : α) in s, ↑↑(↑(Measure.condKernel ρ) a) t ∂Measure.fst ρ = ∫⁻ (x : α), ↑↑(↑(Measure.condKernel ρ) x) {c | x ∈ s ∧ c ∈ t} ∂Measure.fst ρ ** rw [← lintegral_indicator _ hs] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t this : ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) ⊢ ∫⁻ (a : α), indicator s (fun a => ↑↑(↑(Measure.condKernel ρ) a) t) a ∂Measure.fst ρ = ∫⁻ (x : α), ↑↑(↑(Measure.condKernel ρ) x) {c | x ∈ s ∧ c ∈ t} ∂Measure.fst ρ ** congr ** case e_f α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t this : ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) ⊢ (fun a => indicator s (fun a => ↑↑(↑(Measure.condKernel ρ) a) t) a) = fun x => ↑↑(↑(Measure.condKernel ρ) x) {c | x ∈ s ∧ c ∈ t} ** ext1 x ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t ⊢ ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) ** congr ** case e_self.e_self α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t ⊢ ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) () ** exact measure_eq_compProd ρ ** case e_f.h α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t this : ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) x : α ⊢ indicator s (fun a => ↑↑(↑(Measure.condKernel ρ) a) t) x = ↑↑(↑(Measure.condKernel ρ) x) {c | x ∈ s ∧ c ∈ t} ** rw [indicator_apply] ** case e_f.h α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t this : ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) x : α ⊢ (if x ∈ s then ↑↑(↑(Measure.condKernel ρ) x) t else 0) = ↑↑(↑(Measure.condKernel ρ) x) {c | x ∈ s ∧ c ∈ t} ** split_ifs with hx ** case pos α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t this : ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) x : α hx : x ∈ s ⊢ ↑↑(↑(Measure.condKernel ρ) x) t = ↑↑(↑(Measure.condKernel ρ) x) {c | x ∈ s ∧ c ∈ t} ** simp only [hx, if_true, true_and_iff, setOf_mem_eq] ** case neg α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t this : ↑↑ρ (s ×ˢ t) = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) (s ×ˢ t) x : α hx : ¬x ∈ s ⊢ 0 = ↑↑(↑(Measure.condKernel ρ) x) {c | x ∈ s ∧ c ∈ t} ** simp only [hx, if_false, false_and_iff, setOf_false, measure_empty] ** Qed
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ProbabilityTheory.lintegral_condKernel ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), ∫⁻ (ω : Ω), f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (x : α × Ω), f x ∂ρ ** conv_rhs => rw [measure_eq_compProd ρ] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), ∫⁻ (ω : Ω), f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (x : α × Ω), f x ∂↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) () ** rw [kernel.lintegral_compProd _ _ _ hf, kernel.const_apply] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), ∫⁻ (ω : Ω), f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (b : α), ∫⁻ (c : Ω), f (b, c) ∂↑(kernel.prodMkLeft Unit (Measure.condKernel ρ)) ((), b) ∂Measure.fst ρ ** simp_rw [kernel.prodMkLeft_apply] ** Qed
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ProbabilityTheory.set_lintegral_condKernel ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t ⊢ ∫⁻ (a : α) in s, ∫⁻ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (x : α × Ω) in s ×ˢ t, f x ∂ρ ** conv_rhs => rw [measure_eq_compProd ρ] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t ⊢ ∫⁻ (a : α) in s, ∫⁻ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (x : α × Ω) in s ×ˢ t, f x ∂↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) () ** rw [← kernel.restrict_apply _ (hs.prod ht), ← kernel.compProd_restrict hs ht,
kernel.lintegral_compProd _ _ _ hf, kernel.restrict_apply] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t ⊢ ∫⁻ (a : α) in s, ∫⁻ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (b : α) in s, ∫⁻ (c : Ω), f (b, c) ∂↑(kernel.restrict (kernel.prodMkLeft Unit (Measure.condKernel ρ)) ht) ((), b) ∂↑(kernel.const Unit (Measure.fst ρ)) () ** conv_rhs => enter [2, b, 1]; rw [kernel.restrict_apply _ ht] ** Qed
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ProbabilityTheory.set_lintegral_condKernel_univ_right ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α) in s, ∫⁻ (ω : Ω), f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (x : α × Ω) in s ×ˢ univ, f x ∂ρ ** rw [← set_lintegral_condKernel ρ hf hs MeasurableSet.univ] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α) in s, ∫⁻ (ω : Ω), f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (a : α) in s, ∫⁻ (ω : Ω) in univ, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ ** simp_rw [Measure.restrict_univ] ** Qed
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ProbabilityTheory.set_lintegral_condKernel_univ_left ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f t : Set Ω ht : MeasurableSet t ⊢ ∫⁻ (a : α), ∫⁻ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (x : α × Ω) in univ ×ˢ t, f x ∂ρ ** rw [← set_lintegral_condKernel ρ hf MeasurableSet.univ ht] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁵ : TopologicalSpace Ω inst✝⁴ : PolishSpace Ω inst✝³ : MeasurableSpace Ω inst✝² : BorelSpace Ω inst✝¹ : Nonempty Ω ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → ℝ≥0∞ hf : Measurable f t : Set Ω ht : MeasurableSet t ⊢ ∫⁻ (a : α), ∫⁻ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫⁻ (a : α) in univ, ∫⁻ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ ** simp_rw [Measure.restrict_univ] ** Qed
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ProbabilityTheory.integral_condKernel ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf : Integrable f ⊢ ∫ (a : α), ∫ (x : Ω), f (a, x) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (ω : α × Ω), f ω ∂ρ ** conv_rhs => rw [measure_eq_compProd ρ] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf : Integrable f ⊢ ∫ (a : α), ∫ (x : Ω), f (a, x) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (ω : α × Ω), f ω ∂↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) () ** have hf': Integrable f ((kernel.const Unit ρ.fst ⊗ₖ kernel.prodMkLeft Unit ρ.condKernel) ()) := by
rwa [measure_eq_compProd ρ] at hf ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf : Integrable f hf' : Integrable f ⊢ ∫ (a : α), ∫ (x : Ω), f (a, x) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (ω : α × Ω), f ω ∂↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) () ** rw [integral_compProd hf', kernel.const_apply] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf : Integrable f hf' : Integrable f ⊢ ∫ (a : α), ∫ (x : Ω), f (a, x) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (x : α), ∫ (y : Ω), f (x, y) ∂↑(kernel.prodMkLeft Unit (Measure.condKernel ρ)) ((), x) ∂Measure.fst ρ ** simp_rw [kernel.prodMkLeft_apply] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf : Integrable f ⊢ Integrable f ** rwa [measure_eq_compProd ρ] at hf ** Qed
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ProbabilityTheory.set_integral_condKernel ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t hf : IntegrableOn f (s ×ˢ t) ⊢ ∫ (a : α) in s, ∫ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (x : α × Ω) in s ×ˢ t, f x ∂ρ ** conv_rhs => rw [measure_eq_compProd ρ] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t hf : IntegrableOn f (s ×ˢ t) ⊢ ∫ (a : α) in s, ∫ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (x : α × Ω) in s ×ˢ t, f x ∂↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) () ** rw [set_integral_compProd hs ht] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t hf : IntegrableOn f (s ×ˢ t) ⊢ ∫ (a : α) in s, ∫ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (x : α) in s, ∫ (y : Ω) in t, f (x, y) ∂↑(kernel.prodMkLeft Unit (Measure.condKernel ρ)) ((), x) ∂↑(kernel.const Unit (Measure.fst ρ)) () ** simp_rw [kernel.prodMkLeft_apply, kernel.const_apply] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E s : Set α hs : MeasurableSet s t : Set Ω ht : MeasurableSet t hf : IntegrableOn f (s ×ˢ t) ⊢ IntegrableOn (fun x => f x) (s ×ˢ t) ** rwa [measure_eq_compProd ρ] at hf ** Qed
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ProbabilityTheory.set_integral_condKernel_univ_right ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E s : Set α hs : MeasurableSet s hf : IntegrableOn f (s ×ˢ univ) ⊢ ∫ (a : α) in s, ∫ (ω : Ω), f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (x : α × Ω) in s ×ˢ univ, f x ∂ρ ** rw [← set_integral_condKernel hs MeasurableSet.univ hf] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E s : Set α hs : MeasurableSet s hf : IntegrableOn f (s ×ˢ univ) ⊢ ∫ (a : α) in s, ∫ (ω : Ω), f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (a : α) in s, ∫ (ω : Ω) in univ, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ ** simp_rw [Measure.restrict_univ] ** Qed
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ProbabilityTheory.set_integral_condKernel_univ_left ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E t : Set Ω ht : MeasurableSet t hf : IntegrableOn f (univ ×ˢ t) ⊢ ∫ (a : α), ∫ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (x : α × Ω) in univ ×ˢ t, f x ∂ρ ** rw [← set_integral_condKernel MeasurableSet.univ ht hf] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁹ : TopologicalSpace Ω inst✝⁸ : PolishSpace Ω inst✝⁷ : MeasurableSpace Ω inst✝⁶ : BorelSpace Ω inst✝⁵ : Nonempty Ω ρ✝ : Measure (α × Ω) inst✝⁴ : IsFiniteMeasure ρ✝ E : Type u_3 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : CompleteSpace E ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E t : Set Ω ht : MeasurableSet t hf : IntegrableOn f (univ ×ˢ t) ⊢ ∫ (a : α), ∫ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ = ∫ (a : α) in univ, ∫ (ω : Ω) in t, f (a, ω) ∂↑(Measure.condKernel ρ) a ∂Measure.fst ρ ** simp_rw [Measure.restrict_univ] ** Qed
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ProbabilityTheory.eq_condKernel_of_measure_eq_compProd' ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s ⊢ ∀ᵐ (x : α) ∂Measure.fst ρ, ↑↑(↑κ x) s = ↑↑(↑(Measure.condKernel ρ) x) s ** refine' ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite
(kernel.measurable_coe κ hs) (kernel.measurable_coe ρ.condKernel hs) _ ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s ⊢ ∀ (s_1 : Set α), MeasurableSet s_1 → ↑↑(Measure.fst ρ) s_1 < ⊤ → ∫⁻ (x : α) in s_1, ↑↑(↑κ x) s ∂Measure.fst ρ = ∫⁻ (x : α) in s_1, ↑↑(↑(Measure.condKernel ρ) x) s ∂Measure.fst ρ ** intros t ht _ ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s t : Set α ht : MeasurableSet t a✝ : ↑↑(Measure.fst ρ) t < ⊤ ⊢ ∫⁻ (x : α) in t, ↑↑(↑κ x) s ∂Measure.fst ρ = ∫⁻ (x : α) in t, ↑↑(↑(Measure.condKernel ρ) x) s ∂Measure.fst ρ ** conv_rhs => rw [set_lintegral_condKernel_eq_measure_prod _ ht hs, hκ] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s t : Set α ht : MeasurableSet t a✝ : ↑↑(Measure.fst ρ) t < ⊤ ⊢ ∫⁻ (x : α) in t, ↑↑(↑κ x) s ∂Measure.fst ρ = ↑↑(↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) ()) (t ×ˢ s) ** simp only [kernel.compProd_apply _ _ _ (ht.prod hs), kernel.prodMkLeft_apply, Set.mem_prod,
kernel.lintegral_const, ← lintegral_indicator _ ht] ** α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s t : Set α ht : MeasurableSet t a✝ : ↑↑(Measure.fst ρ) t < ⊤ ⊢ ∫⁻ (a : α), indicator t (fun a => ↑↑(↑κ a) s) a ∂Measure.fst ρ = ∫⁻ (x : α), ↑↑(↑κ x) {c | x ∈ t ∧ c ∈ s} ∂Measure.fst ρ ** congr ** case e_f α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s t : Set α ht : MeasurableSet t a✝ : ↑↑(Measure.fst ρ) t < ⊤ ⊢ (fun a => indicator t (fun a => ↑↑(↑κ a) s) a) = fun x => ↑↑(↑κ x) {c | x ∈ t ∧ c ∈ s} ** ext x ** case e_f.h α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s t : Set α ht : MeasurableSet t a✝ : ↑↑(Measure.fst ρ) t < ⊤ x : α ⊢ indicator t (fun a => ↑↑(↑κ a) s) x = ↑↑(↑κ x) {c | x ∈ t ∧ c ∈ s} ** by_cases hx : x ∈ t ** case pos α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s t : Set α ht : MeasurableSet t a✝ : ↑↑(Measure.fst ρ) t < ⊤ x : α hx : x ∈ t ⊢ indicator t (fun a => ↑↑(↑κ a) s) x = ↑↑(↑κ x) {c | x ∈ t ∧ c ∈ s} case neg α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s t : Set α ht : MeasurableSet t a✝ : ↑↑(Measure.fst ρ) t < ⊤ x : α hx : ¬x ∈ t ⊢ indicator t (fun a => ↑↑(↑κ a) s) x = ↑↑(↑κ x) {c | x ∈ t ∧ c ∈ s} ** all_goals simp [hx] ** case neg α : Type u_1 mα : MeasurableSpace α Ω : Type u_2 inst✝⁶ : TopologicalSpace Ω inst✝⁵ : PolishSpace Ω inst✝⁴ : MeasurableSpace Ω inst✝³ : BorelSpace Ω inst✝² : Nonempty Ω ρ : Measure (α × Ω) inst✝¹ : IsFiniteMeasure ρ κ : { x // x ∈ kernel α Ω } inst✝ : IsSFiniteKernel κ hκ : ρ = ↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit κ) () s : Set Ω hs : MeasurableSet s t : Set α ht : MeasurableSet t a✝ : ↑↑(Measure.fst ρ) t < ⊤ x : α hx : ¬x ∈ t ⊢ indicator t (fun a => ↑↑(↑κ a) s) x = ↑↑(↑κ x) {c | x ∈ t ∧ c ∈ s} ** simp [hx] ** Qed
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MeasureTheory.AEStronglyMeasurable.ae_integrable_condKernel_iff ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → F hf : AEStronglyMeasurable f ρ ⊢ ((∀ᵐ (a : α) ∂Measure.fst ρ, Integrable fun ω => f (a, ω)) ∧ Integrable fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂↑(Measure.condKernel ρ) a) ↔ Integrable f ** rw [measure_eq_compProd ρ] at hf ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → F hf : AEStronglyMeasurable f (↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) ⊢ ((∀ᵐ (a : α) ∂Measure.fst ρ, Integrable fun ω => f (a, ω)) ∧ Integrable fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂↑(Measure.condKernel ρ) a) ↔ Integrable f ** rw [integrable_compProd_iff hf] ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → F hf : AEStronglyMeasurable f (↑(kernel.const Unit (Measure.fst ρ) ⊗ₖ kernel.prodMkLeft Unit (Measure.condKernel ρ)) ()) ⊢ ((∀ᵐ (a : α) ∂Measure.fst ρ, Integrable fun ω => f (a, ω)) ∧ Integrable fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂↑(Measure.condKernel ρ) a) ↔ (∀ᵐ (x : α) ∂↑(kernel.const Unit (Measure.fst ρ)) (), Integrable fun y => f (x, y)) ∧ Integrable fun x => ∫ (y : Ω), ‖f (x, y)‖ ∂↑(kernel.prodMkLeft Unit (Measure.condKernel ρ)) ((), x) ** simp_rw [kernel.prodMkLeft_apply, kernel.const_apply] ** Qed
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MeasureTheory.Integrable.condKernel_ae ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → F hf_int : Integrable f ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, Integrable fun ω => f (a, ω) ** have hf_ae : AEStronglyMeasurable f ρ := hf_int.1 ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → F hf_int : Integrable f hf_ae : AEStronglyMeasurable f ρ ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, Integrable fun ω => f (a, ω) ** rw [← hf_ae.ae_integrable_condKernel_iff] at hf_int ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → F hf_int : (∀ᵐ (a : α) ∂Measure.fst ρ, Integrable fun ω => f (a, ω)) ∧ Integrable fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂↑(Measure.condKernel ρ) a hf_ae : AEStronglyMeasurable f ρ ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, Integrable fun ω => f (a, ω) ** exact hf_int.1 ** Qed
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MeasureTheory.Integrable.integral_norm_condKernel ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → F hf_int : Integrable f ⊢ Integrable fun x => ∫ (y : Ω), ‖f (x, y)‖ ∂↑(Measure.condKernel ρ) x ** have hf_ae : AEStronglyMeasurable f ρ := hf_int.1 ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → F hf_int : Integrable f hf_ae : AEStronglyMeasurable f ρ ⊢ Integrable fun x => ∫ (y : Ω), ‖f (x, y)‖ ∂↑(Measure.condKernel ρ) x ** rw [← hf_ae.ae_integrable_condKernel_iff] at hf_int ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → F hf_int : (∀ᵐ (a : α) ∂Measure.fst ρ, Integrable fun ω => f (a, ω)) ∧ Integrable fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂↑(Measure.condKernel ρ) a hf_ae : AEStronglyMeasurable f ρ ⊢ Integrable fun x => ∫ (y : Ω), ‖f (x, y)‖ ∂↑(Measure.condKernel ρ) x ** exact hf_int.2 ** Qed
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MeasureTheory.Integrable.norm_integral_condKernel ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf_int : Integrable f ⊢ Integrable fun x => ‖∫ (y : Ω), f (x, y) ∂↑(Measure.condKernel ρ) x‖ ** refine' hf_int.integral_norm_condKernel.mono hf_int.1.integral_condKernel.norm _ ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf_int : Integrable f ⊢ ∀ᵐ (a : α) ∂Measure.fst ρ, ‖‖∫ (y : Ω), f (a, y) ∂↑(Measure.condKernel ρ) a‖‖ ≤ ‖∫ (y : Ω), ‖f (a, y)‖ ∂↑(Measure.condKernel ρ) a‖ ** refine' eventually_of_forall fun x => _ ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf_int : Integrable f x : α ⊢ ‖‖∫ (y : Ω), f (x, y) ∂↑(Measure.condKernel ρ) x‖‖ ≤ ‖∫ (y : Ω), ‖f (x, y)‖ ∂↑(Measure.condKernel ρ) x‖ ** rw [norm_norm] ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf_int : Integrable f x : α ⊢ ‖∫ (y : Ω), f (x, y) ∂↑(Measure.condKernel ρ) x‖ ≤ ‖∫ (y : Ω), ‖f (x, y)‖ ∂↑(Measure.condKernel ρ) x‖ ** refine' (norm_integral_le_integral_norm _).trans_eq (Real.norm_of_nonneg _).symm ** α : Type u_1 Ω : Type u_2 E : Type u_3 F : Type u_4 mα : MeasurableSpace α inst✝⁹ : MeasurableSpace Ω inst✝⁸ : TopologicalSpace Ω inst✝⁷ : BorelSpace Ω inst✝⁶ : PolishSpace Ω inst✝⁵ : Nonempty Ω inst✝⁴ : NormedAddCommGroup E inst✝³ : NormedSpace ℝ E inst✝² : CompleteSpace E inst✝¹ : NormedAddCommGroup F ρ : Measure (α × Ω) inst✝ : IsFiniteMeasure ρ f : α × Ω → E hf_int : Integrable f x : α ⊢ 0 ≤ ∫ (a : Ω), ‖f (x, a)‖ ∂↑(Measure.condKernel ρ) x ** exact integral_nonneg_of_ae (eventually_of_forall fun y => norm_nonneg _) ** Qed
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MeasureTheory.isStoppingTime_const ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m i j : ι ⊢ MeasurableSet {ω | (fun x => i) ω ≤ j} ** simp only [MeasurableSet.const] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} ⊢ MeasurableSet {ω | τ ω = i} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} ⊢ MeasurableSet ({ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j}) ** refine' (hτ.measurableSet_le i).diff _ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} ⊢ MeasurableSet (⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j}) ** refine' MeasurableSet.biUnion h_countable fun j _ => _ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} j : ι x✝ : j ∈ Set.range τ ⊢ MeasurableSet (⋃ (_ : j < i), {ω | τ ω ≤ j}) ** by_cases hji : j < i ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι ⊢ {ω | τ ω = i} = {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} ** ext1 a ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι a : Ω ⊢ a ∈ {ω | τ ω = i} ↔ a ∈ {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} ** simp only [Set.mem_setOf_eq, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq',
Set.mem_diff, Set.mem_iUnion, exists_prop, not_exists, not_and, not_le] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι a : Ω ⊢ τ a = i ↔ τ a ≤ i ∧ ∀ (x : Ω), τ x < i → ¬τ a ≤ τ x ** constructor <;> intro h ** case h.mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι a : Ω h : τ a = i ⊢ τ a ≤ i ∧ ∀ (x : Ω), τ x < i → ¬τ a ≤ τ x ** simp only [h, lt_iff_le_not_le, le_refl, and_imp, imp_self, imp_true_iff, and_self_iff] ** case h.mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι a : Ω h : τ a ≤ i ∧ ∀ (x : Ω), τ x < i → ¬τ a ≤ τ x ⊢ τ a = i ** have h_lt_or_eq : τ a < i ∨ τ a = i := lt_or_eq_of_le h.1 ** case h.mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι a : Ω h : τ a ≤ i ∧ ∀ (x : Ω), τ x < i → ¬τ a ≤ τ x h_lt_or_eq : τ a < i ∨ τ a = i ⊢ τ a = i ** rcases h_lt_or_eq with (h_lt | rfl) ** case h.mpr.inl Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι a : Ω h : τ a ≤ i ∧ ∀ (x : Ω), τ x < i → ¬τ a ≤ τ x h_lt : τ a < i ⊢ τ a = i ** exfalso ** case h.mpr.inl.h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι a : Ω h : τ a ≤ i ∧ ∀ (x : Ω), τ x < i → ¬τ a ≤ τ x h_lt : τ a < i ⊢ False ** exact h.2 a h_lt (le_refl (τ a)) ** case h.mpr.inr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) a : Ω h : τ a ≤ τ a ∧ ∀ (x : Ω), τ x < τ a → ¬τ a ≤ τ x ⊢ τ a = τ a ** rfl ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} j : ι x✝ : j ∈ Set.range τ hji : j < i ⊢ MeasurableSet (⋃ (_ : j < i), {ω | τ ω ≤ j}) ** simp only [hji, Set.iUnion_true] ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} j : ι x✝ : j ∈ Set.range τ hji : j < i ⊢ MeasurableSet {ω | τ ω ≤ j} ** exact f.mono hji.le _ (hτ.measurableSet_le j) ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} j : ι x✝ : j ∈ Set.range τ hji : ¬j < i ⊢ MeasurableSet (⋃ (_ : j < i), {ω | τ ω ≤ j}) ** simp only [hji, Set.iUnion_false] ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} \ ⋃ j ∈ Set.range τ, ⋃ (_ : j < i), {ω | τ ω ≤ j} j : ι x✝ : j ∈ Set.range τ hji : ¬j < i ⊢ MeasurableSet ∅ ** exact @MeasurableSet.empty _ (f i) ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι ⊢ MeasurableSet {ω | τ ω < i} ** have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} := by ext1 ω; simp [lt_iff_le_and_ne] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} ⊢ MeasurableSet {ω | τ ω < i} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} ⊢ MeasurableSet ({ω | τ ω ≤ i} \ {ω | τ ω = i}) ** exact (hτ.measurableSet_le i).diff (hτ.measurableSet_eq_of_countable_range h_countable i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι ⊢ {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : PartialOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι ω : Ω ⊢ ω ∈ {ω | τ ω < i} ↔ ω ∈ {ω | τ ω ≤ i} \ {ω | τ ω = i} ** simp [lt_iff_le_and_ne] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range ** Ω : Type u_1 β : Type u_2 ι✝ : Type u_3 m : MeasurableSpace Ω inst✝¹ : PartialOrder ι✝ τ✝ : Ω → ι✝ f✝ : Filtration ι✝ m ι : Type u_4 inst✝ : LinearOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι ⊢ MeasurableSet {ω | i ≤ τ ω} ** have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ := by
ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt] ** Ω : Type u_1 β : Type u_2 ι✝ : Type u_3 m : MeasurableSpace Ω inst✝¹ : PartialOrder ι✝ τ✝ : Ω → ι✝ f✝ : Filtration ι✝ m ι : Type u_4 inst✝ : LinearOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ ⊢ MeasurableSet {ω | i ≤ τ ω} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι✝ : Type u_3 m : MeasurableSpace Ω inst✝¹ : PartialOrder ι✝ τ✝ : Ω → ι✝ f✝ : Filtration ι✝ m ι : Type u_4 inst✝ : LinearOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι this : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ ⊢ MeasurableSet {ω | τ ω < i}ᶜ ** exact (hτ.measurableSet_lt_of_countable_range h_countable i).compl ** Ω : Type u_1 β : Type u_2 ι✝ : Type u_3 m : MeasurableSpace Ω inst✝¹ : PartialOrder ι✝ τ✝ : Ω → ι✝ f✝ : Filtration ι✝ m ι : Type u_4 inst✝ : LinearOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι ⊢ {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι✝ : Type u_3 m : MeasurableSpace Ω inst✝¹ : PartialOrder ι✝ τ✝ : Ω → ι✝ f✝ : Filtration ι✝ m ι : Type u_4 inst✝ : LinearOrder ι τ : Ω → ι f : Filtration ι m hτ : IsStoppingTime f τ h_countable : Set.Countable (Set.range τ) i : ι ω : Ω ⊢ ω ∈ {ω | i ≤ τ ω} ↔ ω ∈ {ω | τ ω < i}ᶜ ** simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_gt ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ : Ω → ι hτ : IsStoppingTime f τ i : ι ⊢ MeasurableSet {ω | i < τ ω} ** have : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ := by
ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ : Ω → ι hτ : IsStoppingTime f τ i : ι this : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ ⊢ MeasurableSet {ω | i < τ ω} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ : Ω → ι hτ : IsStoppingTime f τ i : ι this : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ ⊢ MeasurableSet {ω | τ ω ≤ i}ᶜ ** exact (hτ.measurableSet_le i).compl ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ : Ω → ι hτ : IsStoppingTime f τ i : ι ⊢ {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : LinearOrder ι f : Filtration ι m τ : Ω → ι hτ : IsStoppingTime f τ i : ι ω : Ω ⊢ ω ∈ {ω | i < τ ω} ↔ ω ∈ {ω | τ ω ≤ i}ᶜ ** simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_lt_of_isLUB ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i ⊢ MeasurableSet {ω | τ ω < i} ** by_cases hi_min : IsMin i ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i ⊢ MeasurableSet {ω | τ ω < i} ** obtain ⟨seq, -, -, h_tendsto, h_bound⟩ :
∃ seq : ℕ → ι, Monotone seq ∧ (∀ j, seq j ≤ i) ∧ Tendsto seq atTop (𝓝 i) ∧ ∀ j, seq j < i ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i ⊢ ∃ seq, Monotone seq ∧ (∀ (j : ℕ), seq j ≤ i) ∧ Tendsto seq atTop (𝓝 i) ∧ ∀ (j : ℕ), seq j < i case neg.intro.intro.intro.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i ⊢ MeasurableSet {ω | τ ω < i} ** exact h_lub.exists_seq_monotone_tendsto (not_isMin_iff.mp hi_min) ** case neg.intro.intro.intro.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i h_Ioi_eq_Union : Set.Iio i = ⋃ j, {k | k ≤ seq j} ⊢ MeasurableSet {ω | τ ω < i} ** have h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i := by
ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_preimage, Set.mem_Iio] ** case neg.intro.intro.intro.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i h_Ioi_eq_Union : Set.Iio i = ⋃ j, {k | k ≤ seq j} h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i ⊢ MeasurableSet {ω | τ ω < i} ** rw [h_lt_eq_preimage, h_Ioi_eq_Union] ** case neg.intro.intro.intro.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i h_Ioi_eq_Union : Set.Iio i = ⋃ j, {k | k ≤ seq j} h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i ⊢ MeasurableSet (τ ⁻¹' ⋃ j, {k | k ≤ seq j}) ** simp only [Set.preimage_iUnion, Set.preimage_setOf_eq] ** case neg.intro.intro.intro.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i h_Ioi_eq_Union : Set.Iio i = ⋃ j, {k | k ≤ seq j} h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i ⊢ MeasurableSet (⋃ i, {a | τ a ≤ seq i}) ** exact MeasurableSet.iUnion fun n => f.mono (h_bound n).le _ (hτ.measurableSet_le (seq n)) ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : IsMin i ⊢ MeasurableSet {ω | τ ω < i} ** suffices {ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i) ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : IsMin i ⊢ {ω | τ ω < i} = ∅ ** ext1 ω ** case pos.h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : IsMin i ω : Ω ⊢ ω ∈ {ω | τ ω < i} ↔ ω ∈ ∅ ** simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff] ** case pos.h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : IsMin i ω : Ω ⊢ ¬τ ω < i ** exact isMin_iff_forall_not_lt.mp hi_min (τ ω) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : IsMin i this : {ω | τ ω < i} = ∅ ⊢ MeasurableSet {ω | τ ω < i} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : IsMin i this : {ω | τ ω < i} = ∅ ⊢ MeasurableSet ∅ ** exact @MeasurableSet.empty _ (f i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i ⊢ Set.Iio i = ⋃ j, {k | k ≤ seq j} ** ext1 k ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i k : ι ⊢ k ∈ Set.Iio i ↔ k ∈ ⋃ j, {k | k ≤ seq j} ** simp only [Set.mem_Iio, Set.mem_iUnion, Set.mem_setOf_eq] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i k : ι ⊢ k < i ↔ ∃ i, k ≤ seq i ** refine' ⟨fun hk_lt_i => _, fun h_exists_k_le_seq => _⟩ ** case h.refine'_1 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i k : ι hk_lt_i : k < i ⊢ ∃ i, k ≤ seq i ** rw [tendsto_atTop'] at h_tendsto ** case h.refine'_1 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : ∀ (s : Set ι), s ∈ 𝓝 i → ∃ a, ∀ (b : ℕ), b ≥ a → seq b ∈ s h_bound : ∀ (j : ℕ), seq j < i k : ι hk_lt_i : k < i ⊢ ∃ i, k ≤ seq i ** have h_nhds : Set.Ici k ∈ 𝓝 i :=
mem_nhds_iff.mpr ⟨Set.Ioi k, Set.Ioi_subset_Ici le_rfl, isOpen_Ioi, hk_lt_i⟩ ** case h.refine'_1 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : ∀ (s : Set ι), s ∈ 𝓝 i → ∃ a, ∀ (b : ℕ), b ≥ a → seq b ∈ s h_bound : ∀ (j : ℕ), seq j < i k : ι hk_lt_i : k < i h_nhds : Set.Ici k ∈ 𝓝 i ⊢ ∃ i, k ≤ seq i ** obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, b ≥ a → k ≤ seq b := h_tendsto (Set.Ici k) h_nhds ** case h.refine'_1.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : ∀ (s : Set ι), s ∈ 𝓝 i → ∃ a, ∀ (b : ℕ), b ≥ a → seq b ∈ s h_bound : ∀ (j : ℕ), seq j < i k : ι hk_lt_i : k < i h_nhds : Set.Ici k ∈ 𝓝 i a : ℕ ha : ∀ (b : ℕ), b ≥ a → k ≤ seq b ⊢ ∃ i, k ≤ seq i ** exact ⟨a, ha a le_rfl⟩ ** case h.refine'_2 Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i k : ι h_exists_k_le_seq : ∃ i, k ≤ seq i ⊢ k < i ** obtain ⟨j, hk_seq_j⟩ := h_exists_k_le_seq ** case h.refine'_2.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i k : ι j : ℕ hk_seq_j : k ≤ seq j ⊢ k < i ** exact hk_seq_j.trans_lt (h_bound j) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i h_Ioi_eq_Union : Set.Iio i = ⋃ j, {k | k ≤ seq j} ⊢ {ω | τ ω < i} = τ ⁻¹' Set.Iio i ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι h_lub : IsLUB (Set.Iio i) i hi_min : ¬IsMin i seq : ℕ → ι h_tendsto : Tendsto seq atTop (𝓝 i) h_bound : ∀ (j : ℕ), seq j < i h_Ioi_eq_Union : Set.Iio i = ⋃ j, {k | k ≤ seq j} ω : Ω ⊢ ω ∈ {ω | τ ω < i} ↔ ω ∈ τ ⁻¹' Set.Iio i ** simp only [Set.mem_setOf_eq, Set.mem_preimage, Set.mem_Iio] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_lt ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ MeasurableSet {ω | τ ω < i} ** obtain ⟨i', hi'_lub⟩ : ∃ i', IsLUB (Set.Iio i) i' ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ ∃ i', IsLUB (Set.Iio i) i' case intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i i' : ι hi'_lub : IsLUB (Set.Iio i) i' ⊢ MeasurableSet {ω | τ ω < i} ** exact exists_lub_Iio i ** case intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i i' : ι hi'_lub : IsLUB (Set.Iio i) i' ⊢ MeasurableSet {ω | τ ω < i} ** cases' lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i h_Iio_eq_Iic ** case intro.inl Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i i' : ι hi'_lub : IsLUB (Set.Iio i) i' hi'_eq_i : i' = i ⊢ MeasurableSet {ω | τ ω < i} ** rw [← hi'_eq_i] at hi'_lub ⊢ ** case intro.inl Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i i' : ι hi'_lub : IsLUB (Set.Iio i') i' hi'_eq_i : i' = i ⊢ MeasurableSet {ω | τ ω < i'} ** exact hτ.measurableSet_lt_of_isLUB i' hi'_lub ** case intro.inr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i i' : ι hi'_lub : IsLUB (Set.Iio i) i' h_Iio_eq_Iic : Set.Iio i = Set.Iic i' ⊢ MeasurableSet {ω | τ ω < i} ** have h_lt_eq_preimage : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iio i := rfl ** case intro.inr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i i' : ι hi'_lub : IsLUB (Set.Iio i) i' h_Iio_eq_Iic : Set.Iio i = Set.Iic i' h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i ⊢ MeasurableSet {ω | τ ω < i} ** rw [h_lt_eq_preimage, h_Iio_eq_Iic] ** case intro.inr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i i' : ι hi'_lub : IsLUB (Set.Iio i) i' h_Iio_eq_Iic : Set.Iio i = Set.Iic i' h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i ⊢ MeasurableSet (τ ⁻¹' Set.Iic i') ** exact f.mono (lub_Iio_le i hi'_lub) _ (hτ.measurableSet_le i') ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_ge ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ MeasurableSet {ω | i ≤ τ ω} ** have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ := by
ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι this : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ ⊢ MeasurableSet {ω | i ≤ τ ω} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι this : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ ⊢ MeasurableSet {ω | τ ω < i}ᶜ ** exact (hτ.measurableSet_lt i).compl ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ω : Ω ⊢ ω ∈ {ω | i ≤ τ ω} ↔ ω ∈ {ω | τ ω < i}ᶜ ** simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_eq ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ MeasurableSet {ω | τ ω = i} ** have : {ω | τ ω = i} = {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i} := by
ext1 ω; simp only [Set.mem_setOf_eq, ge_iff_le, Set.mem_inter_iff, le_antisymm_iff] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i} ⊢ MeasurableSet {ω | τ ω = i} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι this : {ω | τ ω = i} = {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i} ⊢ MeasurableSet ({ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i}) ** exact (hτ.measurableSet_le i).inter (hτ.measurableSet_ge i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ {ω | τ ω = i} = {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i} ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ω : Ω ⊢ ω ∈ {ω | τ ω = i} ↔ ω ∈ {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i} ** simp only [Set.mem_setOf_eq, ge_iff_le, Set.mem_inter_iff, le_antisymm_iff] ** Qed
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MeasureTheory.isStoppingTime_of_measurableSet_eq ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι inst✝ : Countable ι f : Filtration ι m τ : Ω → ι hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i} ⊢ IsStoppingTime f τ ** intro i ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι inst✝ : Countable ι f : Filtration ι m τ : Ω → ι hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i} i : ι ⊢ MeasurableSet {ω | τ ω ≤ i} ** rw [show {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k} by ext; simp] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι inst✝ : Countable ι f : Filtration ι m τ : Ω → ι hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i} i : ι ⊢ MeasurableSet (⋃ k, ⋃ (_ : k ≤ i), {ω | τ ω = k}) ** refine' MeasurableSet.biUnion (Set.to_countable _) fun k hk => _ ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι inst✝ : Countable ι f : Filtration ι m τ : Ω → ι hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i} i k : ι hk : k ∈ fun k => Preorder.toLE.1 k i ⊢ MeasurableSet {ω | τ ω = k} ** exact f.mono hk _ (hτ k) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι inst✝ : Countable ι f : Filtration ι m τ : Ω → ι hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i} i : ι ⊢ {ω | τ ω ≤ i} = ⋃ k, ⋃ (_ : k ≤ i), {ω | τ ω = k} ** ext ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι inst✝ : Countable ι f : Filtration ι m τ : Ω → ι hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i} i : ι x✝ : Ω ⊢ x✝ ∈ {ω | τ ω ≤ i} ↔ x✝ ∈ ⋃ k, ⋃ (_ : k ≤ i), {ω | τ ω = k} ** simp ** Qed
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MeasureTheory.IsStoppingTime.max ** 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 ⊢ IsStoppingTime f fun ω => max (τ ω) (π ω) ** intro 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 hπ : IsStoppingTime f π i : ι ⊢ MeasurableSet {ω | (fun ω => max (τ ω) (π ω)) ω ≤ i} ** simp_rw [max_le_iff, Set.setOf_and] ** 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 i : ι ⊢ MeasurableSet ({a | τ a ≤ i} ∩ {a | π a ≤ i}) ** exact (hτ i).inter (hπ i) ** Qed
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MeasureTheory.IsStoppingTime.min ** 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 ⊢ IsStoppingTime f fun ω => min (τ ω) (π ω) ** intro 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 hπ : IsStoppingTime f π i : ι ⊢ MeasurableSet {ω | (fun ω => min (τ ω) (π ω)) ω ≤ i} ** simp_rw [min_le_iff, Set.setOf_or] ** 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 i : ι ⊢ MeasurableSet ({a | τ a ≤ i} ∪ {a | π a ≤ i}) ** exact (hτ i).union (hπ i) ** Qed
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MeasureTheory.IsStoppingTime.add_const ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : AddGroup ι inst✝² : Preorder ι inst✝¹ : CovariantClass ι ι (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 inst✝ : CovariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 f : Filtration ι m τ : Ω → ι hτ : IsStoppingTime f τ i : ι hi : 0 ≤ i ⊢ IsStoppingTime f fun ω => τ ω + i ** intro j ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : AddGroup ι inst✝² : Preorder ι inst✝¹ : CovariantClass ι ι (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 inst✝ : CovariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 f : Filtration ι m τ : Ω → ι hτ : IsStoppingTime f τ i : ι hi : 0 ≤ i j : ι ⊢ MeasurableSet {ω | (fun ω => τ ω + i) ω ≤ j} ** simp_rw [← le_sub_iff_add_le] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : AddGroup ι inst✝² : Preorder ι inst✝¹ : CovariantClass ι ι (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 inst✝ : CovariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 f : Filtration ι m τ : Ω → ι hτ : IsStoppingTime f τ i : ι hi : 0 ≤ i j : ι ⊢ MeasurableSet {ω | τ ω ≤ j - i} ** exact f.mono (sub_le_self j hi) _ (hτ (j - i)) ** Qed
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MeasureTheory.IsStoppingTime.add ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m τ π : Ω → ℕ hτ : IsStoppingTime f τ hπ : IsStoppingTime f π ⊢ IsStoppingTime f (τ + π) ** intro i ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m τ π : Ω → ℕ hτ : IsStoppingTime f τ hπ : IsStoppingTime f π i : ℕ ⊢ MeasurableSet {ω | (τ + π) ω ≤ i} ** rw [(_ : {ω | (τ + π) ω ≤ i} = ⋃ k ≤ i, {ω | π ω = k} ∩ {ω | τ ω + k ≤ i})] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m τ π : Ω → ℕ hτ : IsStoppingTime f τ hπ : IsStoppingTime f π i : ℕ ⊢ {ω | (τ + π) ω ≤ i} = ⋃ k, ⋃ (_ : k ≤ i), {ω | π ω = k} ∩ {ω | τ ω + k ≤ i} ** ext ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m τ π : Ω → ℕ hτ : IsStoppingTime f τ hπ : IsStoppingTime f π i : ℕ ω : Ω ⊢ ω ∈ {ω | (τ + π) ω ≤ i} ↔ ω ∈ ⋃ k, ⋃ (_ : k ≤ i), {ω | π ω = k} ∩ {ω | τ ω + k ≤ i} ** simp only [Pi.add_apply, Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_prop] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m τ π : Ω → ℕ hτ : IsStoppingTime f τ hπ : IsStoppingTime f π i : ℕ ω : Ω ⊢ τ ω + π ω ≤ i ↔ ∃ i_1, i_1 ≤ i ∧ π ω = i_1 ∧ τ ω + i_1 ≤ i ** refine' ⟨fun h => ⟨π ω, by linarith, rfl, h⟩, _⟩ ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m τ π : Ω → ℕ hτ : IsStoppingTime f τ hπ : IsStoppingTime f π i : ℕ ω : Ω ⊢ (∃ i_1, i_1 ≤ i ∧ π ω = i_1 ∧ τ ω + i_1 ≤ i) → τ ω + π ω ≤ i ** rintro ⟨j, hj, rfl, h⟩ ** case h.intro.intro.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m τ π : Ω → ℕ hτ : IsStoppingTime f τ hπ : IsStoppingTime f π i : ℕ ω : Ω hj : π ω ≤ i h : τ ω + π ω ≤ i ⊢ τ ω + π ω ≤ i ** assumption ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m τ π : Ω → ℕ hτ : IsStoppingTime f τ hπ : IsStoppingTime f π i : ℕ ⊢ MeasurableSet (⋃ k, ⋃ (_ : k ≤ i), {ω | π ω = k} ∩ {ω | τ ω + k ≤ i}) ** exact MeasurableSet.iUnion fun k =>
MeasurableSet.iUnion fun hk => (hπ.measurableSet_eq_le hk).inter (hτ.add_const_nat i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω f : Filtration ℕ m τ π : Ω → ℕ hτ : IsStoppingTime f τ hπ : IsStoppingTime f π i : ℕ ω : Ω h : τ ω + π ω ≤ i ⊢ π ω ≤ i ** linarith ** Qed
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MeasureTheory.IsStoppingTime.measurableSpace_mono ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π hle : τ ≤ π ⊢ IsStoppingTime.measurableSpace hτ ≤ IsStoppingTime.measurableSpace hπ ** intro s hs i ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π hle : τ ≤ π s : Set Ω hs : MeasurableSet s i : ι ⊢ MeasurableSet (s ∩ {ω | π ω ≤ i}) ** rw [(_ : s ∩ {ω | π ω ≤ i} = s ∩ {ω | τ ω ≤ i} ∩ {ω | π ω ≤ i})] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π hle : τ ≤ π s : Set Ω hs : MeasurableSet s i : ι ⊢ MeasurableSet (s ∩ {ω | τ ω ≤ i} ∩ {ω | π ω ≤ i}) ** exact (hs i).inter (hπ i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π hle : τ ≤ π s : Set Ω hs : MeasurableSet s i : ι ⊢ s ∩ {ω | π ω ≤ i} = s ∩ {ω | τ ω ≤ i} ∩ {ω | π ω ≤ i} ** ext ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π hle : τ ≤ π s : Set Ω hs : MeasurableSet s i : ι x✝ : Ω ⊢ x✝ ∈ s ∩ {ω | π ω ≤ i} ↔ x✝ ∈ s ∩ {ω | τ ω ≤ i} ∩ {ω | π ω ≤ i} ** simp only [Set.mem_inter_iff, iff_self_and, and_congr_left_iff, Set.mem_setOf_eq] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π hle : τ ≤ π s : Set Ω hs : MeasurableSet s i : ι x✝ : Ω ⊢ π x✝ ≤ i → x✝ ∈ s → τ x✝ ≤ i ** intro hle' _ ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ hπ : IsStoppingTime f π hle : τ ≤ π s : Set Ω hs : MeasurableSet s i : ι x✝ : Ω hle' : π x✝ ≤ i a✝ : x✝ ∈ s ⊢ τ x✝ ≤ i ** exact le_trans (hle _) hle' ** Qed
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MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝ : Countable ι hτ : IsStoppingTime f τ ⊢ IsStoppingTime.measurableSpace hτ ≤ m ** intro s hs ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝ : Countable ι hτ : IsStoppingTime f τ s : Set Ω hs : MeasurableSet s ⊢ MeasurableSet s ** change ∀ i, MeasurableSet[f i] (s ∩ {ω | τ ω ≤ i}) at hs ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝ : Countable ι hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ⊢ MeasurableSet s ** rw [(_ : s = ⋃ i, s ∩ {ω | τ ω ≤ i})] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝ : Countable ι hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ⊢ MeasurableSet (⋃ i, s ∩ {ω | τ ω ≤ i}) ** exact MeasurableSet.iUnion fun i => f.le i _ (hs i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝ : Countable ι hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ⊢ s = ⋃ i, s ∩ {ω | τ ω ≤ i} ** ext ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝ : Countable ι hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω ⊢ ω ∈ s ↔ ω ∈ ⋃ i, s ∩ {ω | τ ω ≤ i} ** constructor <;> rw [Set.mem_iUnion] ** case h.mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝ : Countable ι hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω ⊢ ω ∈ s → ∃ i, ω ∈ s ∩ {ω | τ ω ≤ i} ** exact fun hx => ⟨τ ω, hx, le_rfl⟩ ** case h.mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝ : Countable ι hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω ⊢ (∃ i, ω ∈ s ∩ {ω | τ ω ≤ i}) → ω ∈ s ** rintro ⟨_, hx, _⟩ ** case h.mpr.intro.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝¹ : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝ : Countable ι hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω w✝ : ι hx : ω ∈ s right✝ : ω ∈ {ω | τ ω ≤ w✝} ⊢ ω ∈ s ** exact hx ** Qed
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MeasureTheory.IsStoppingTime.measurableSpace_le' ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ ⊢ IsStoppingTime.measurableSpace hτ ≤ m ** intro s hs ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : MeasurableSet s ⊢ MeasurableSet s ** change ∀ i, MeasurableSet[f i] (s ∩ {ω | τ ω ≤ i}) at hs ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ⊢ MeasurableSet s ** obtain ⟨seq : ℕ → ι, h_seq_tendsto⟩ := (atTop : Filter ι).exists_seq_tendsto ** case intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ⊢ MeasurableSet s ** rw [(_ : s = ⋃ n, s ∩ {ω | τ ω ≤ seq n})] ** case intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ⊢ MeasurableSet (⋃ n, s ∩ {ω | τ ω ≤ seq n}) ** exact MeasurableSet.iUnion fun i => f.le (seq i) _ (hs (seq i)) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ⊢ s = ⋃ n, s ∩ {ω | τ ω ≤ seq n} ** ext ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ω : Ω ⊢ ω ∈ s ↔ ω ∈ ⋃ n, s ∩ {ω | τ ω ≤ seq n} ** constructor <;> rw [Set.mem_iUnion] ** case h.mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ω : Ω ⊢ ω ∈ s → ∃ i, ω ∈ s ∩ {ω | τ ω ≤ seq i} ** intro hx ** case h.mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ω : Ω hx : ω ∈ s ⊢ ∃ i, ω ∈ s ∩ {ω | τ ω ≤ seq i} ** suffices : ∃ i, τ ω ≤ seq i ** case h.mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ω : Ω hx : ω ∈ s this : ∃ i, τ ω ≤ seq i ⊢ ∃ i, ω ∈ s ∩ {ω | τ ω ≤ seq i} case this Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ω : Ω hx : ω ∈ s ⊢ ∃ i, τ ω ≤ seq i ** exact ⟨this.choose, hx, this.choose_spec⟩ ** case this Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ω : Ω hx : ω ∈ s ⊢ ∃ i, τ ω ≤ seq i ** rw [tendsto_atTop] at h_seq_tendsto ** case this Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : ∀ (b : ι), ∀ᶠ (a : ℕ) in atTop, b ≤ seq a ω : Ω hx : ω ∈ s ⊢ ∃ i, τ ω ≤ seq i ** exact (h_seq_tendsto (τ ω)).exists ** case h.mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ω : Ω ⊢ (∃ i, ω ∈ s ∩ {ω | τ ω ≤ seq i}) → ω ∈ s ** rintro ⟨_, hx, _⟩ ** case h.mpr.intro.intro Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝² : Preorder ι f : Filtration ι m τ π : Ω → ι inst✝¹ : IsCountablyGenerated atTop inst✝ : NeBot atTop hτ : IsStoppingTime f τ s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) seq : ℕ → ι h_seq_tendsto : Tendsto seq atTop atTop ω : Ω w✝ : ℕ hx : ω ∈ s right✝ : ω ∈ {ω | τ ω ≤ seq w✝} ⊢ ω ∈ s ** exact hx ** Qed
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MeasureTheory.IsStoppingTime.measurableSpace_const ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι ⊢ IsStoppingTime.measurableSpace (_ : IsStoppingTime f fun x => i) = ↑f i ** ext1 s ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι s : Set Ω ⊢ MeasurableSet s ↔ MeasurableSet s ** rw [IsStoppingTime.measurableSet] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι s : Set Ω ⊢ (∀ (i_1 : ι), MeasurableSet (s ∩ {ω | i ≤ i_1})) ↔ MeasurableSet s ** constructor <;> intro h ** case h.mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι s : Set Ω h : ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | i ≤ i_1}) ⊢ MeasurableSet s ** specialize h i ** case h.mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι s : Set Ω h : MeasurableSet (s ∩ {ω | i ≤ i}) ⊢ MeasurableSet s ** simpa only [le_refl, Set.setOf_true, Set.inter_univ] using h ** case h.mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι s : Set Ω h : MeasurableSet s ⊢ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | i ≤ i_1}) ** intro j ** case h.mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι s : Set Ω h : MeasurableSet s j : ι ⊢ MeasurableSet (s ∩ {ω | i ≤ j}) ** by_cases hij : i ≤ j ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι s : Set Ω h : MeasurableSet s j : ι hij : i ≤ j ⊢ MeasurableSet (s ∩ {ω | i ≤ j}) ** simp only [hij, Set.setOf_true, Set.inter_univ] ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι s : Set Ω h : MeasurableSet s j : ι hij : i ≤ j ⊢ MeasurableSet s ** exact f.mono hij _ h ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f✝ : Filtration ι m τ π : Ω → ι f : Filtration ι m i : ι s : Set Ω h : MeasurableSet s j : ι hij : ¬i ≤ j ⊢ MeasurableSet (s ∩ {ω | i ≤ j}) ** simp only [hij, Set.setOf_false, Set.inter_empty, @MeasurableSet.empty _ (f.1 j)] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_inter_eq_iff ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder 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}) ** have : ∀ j, {ω : Ω | τ ω = i} ∩ {ω : Ω | τ ω ≤ j} = {ω : Ω | τ ω = i} ∩ {_ω | i ≤ j} := by
intro j
ext1 ω
simp only [Set.mem_inter_iff, Set.mem_setOf_eq, and_congr_right_iff]
intro hxi
rw [hxi] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} ⊢ MeasurableSet (s ∩ {ω | τ ω = i}) ↔ MeasurableSet (s ∩ {ω | τ ω = i}) ** constructor <;> intro h ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι ⊢ ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} ** intro j ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i j : ι ⊢ {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i j : ι ω : Ω ⊢ ω ∈ {ω | τ ω = i} ∩ {ω | τ ω ≤ j} ↔ ω ∈ {ω | τ ω = i} ∩ {_ω | i ≤ j} ** simp only [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✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i j : ι ω : Ω ⊢ τ ω = i → (τ ω ≤ j ↔ i ≤ j) ** intro hxi ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i j : ι ω : Ω hxi : τ ω = i ⊢ τ ω ≤ j ↔ i ≤ j ** rw [hxi] ** case mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} h : MeasurableSet (s ∩ {ω | τ ω = i}) ⊢ MeasurableSet (s ∩ {ω | τ ω = i}) ** specialize h i ** case mp Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} h : MeasurableSet (s ∩ {ω | τ ω = i} ∩ {ω | τ ω ≤ i}) ⊢ MeasurableSet (s ∩ {ω | τ ω = i}) ** simpa only [Set.inter_assoc, this, le_refl, Set.setOf_true, Set.inter_univ] using h ** case mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} h : MeasurableSet (s ∩ {ω | τ ω = i}) ⊢ MeasurableSet (s ∩ {ω | τ ω = i}) ** intro j ** case mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} h : MeasurableSet (s ∩ {ω | τ ω = i}) j : ι ⊢ MeasurableSet (s ∩ {ω | τ ω = i} ∩ {ω | τ ω ≤ j}) ** rw [Set.inter_assoc, this] ** case mpr Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} h : MeasurableSet (s ∩ {ω | τ ω = i}) j : ι ⊢ MeasurableSet (s ∩ ({ω | τ ω = i} ∩ {_ω | i ≤ j})) ** by_cases hij : i ≤ j ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} h : MeasurableSet (s ∩ {ω | τ ω = i}) j : ι hij : i ≤ j ⊢ MeasurableSet (s ∩ ({ω | τ ω = i} ∩ {_ω | i ≤ j})) ** simp only [hij, Set.setOf_true, Set.inter_univ] ** case pos Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} h : MeasurableSet (s ∩ {ω | τ ω = i}) j : ι hij : i ≤ j ⊢ MeasurableSet (s ∩ {ω | τ ω = i}) ** exact f.mono hij _ h ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} h : MeasurableSet (s ∩ {ω | τ ω = i}) j : ι hij : ¬i ≤ j ⊢ MeasurableSet (s ∩ ({ω | τ ω = i} ∩ {_ω | i ≤ j})) ** simp [hij] ** case neg Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝ : Preorder ι f : Filtration ι m τ π : Ω → ι hτ : IsStoppingTime f τ s : Set Ω i : ι this : ∀ (j : ι), {ω | τ ω = i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = i} ∩ {_ω | i ≤ j} h : MeasurableSet (s ∩ {ω | τ ω = i}) j : ι hij : ¬i ≤ j ⊢ MeasurableSet ∅ ** convert @MeasurableSet.empty _ (Filtration.seq f j) ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_le' ** 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 ⊢ MeasurableSet {ω | τ ω ≤ i} ** intro 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 i j : ι ⊢ MeasurableSet ({ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j}) ** have : {ω : Ω | τ ω ≤ i} ∩ {ω : Ω | τ ω ≤ j} = {ω : Ω | τ ω ≤ min i j} := by
ext1 ω; simp only [Set.mem_inter_iff, Set.mem_setOf_eq, le_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 i j : ι this : {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j} = {ω | τ ω ≤ min i j} ⊢ MeasurableSet ({ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j}) ** rw [this] ** 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 j : ι this : {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j} = {ω | τ ω ≤ min i j} ⊢ MeasurableSet {ω | τ ω ≤ min i j} ** exact f.mono (min_le_right i j) _ (hτ _) ** 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 j : ι ⊢ {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j} = {ω | τ ω ≤ min i j} ** ext1 ω ** case h 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 j : ι ω : Ω ⊢ ω ∈ {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ j} ↔ ω ∈ {ω | τ ω ≤ min i j} ** simp only [Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_gt' ** 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 ⊢ MeasurableSet {ω | i < τ ω} ** have : {ω : Ω | i < τ ω} = {ω : Ω | τ ω ≤ i}ᶜ := by ext1 ω; simp ** 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 this : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ ⊢ MeasurableSet {ω | i < τ ω} ** rw [this] ** 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 this : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ ⊢ MeasurableSet {ω | τ ω ≤ i}ᶜ ** exact (hτ.measurableSet_le' i).compl ** 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 ⊢ {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ ** ext1 ω ** case h 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 ω : Ω ⊢ ω ∈ {ω | i < τ ω} ↔ ω ∈ {ω | τ ω ≤ i}ᶜ ** simp ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_eq' ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ MeasurableSet {ω | τ ω = i} ** exact hτ.measurableSet_eq i ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_ge' ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ MeasurableSet {ω | i ≤ τ ω} ** have : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω} := by
ext1 ω
simp only [le_iff_lt_or_eq, Set.mem_setOf_eq, Set.mem_union]
rw [@eq_comm _ i, or_comm] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι this : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω} ⊢ MeasurableSet {ω | i ≤ τ ω} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι this : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω} ⊢ MeasurableSet ({ω | τ ω = i} ∪ {ω | i < τ ω}) ** exact (hτ.measurableSet_eq' i).union (hτ.measurableSet_gt' i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω} ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ω : Ω ⊢ ω ∈ {ω | i ≤ τ ω} ↔ ω ∈ {ω | τ ω = i} ∪ {ω | i < τ ω} ** simp only [le_iff_lt_or_eq, Set.mem_setOf_eq, Set.mem_union] ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ω : Ω ⊢ i < τ ω ∨ i = τ ω ↔ τ ω = i ∨ i < τ ω ** rw [@eq_comm _ i, or_comm] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_lt' ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ MeasurableSet {ω | τ ω < i} ** have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} := by
ext1 ω
simp only [lt_iff_le_and_ne, Set.mem_setOf_eq, Set.mem_diff] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} ⊢ MeasurableSet {ω | τ ω < i} ** rw [this] ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} ⊢ MeasurableSet ({ω | τ ω ≤ i} \ {ω | τ ω = i}) ** exact (hτ.measurableSet_le' i).diff (hτ.measurableSet_eq' i) ** Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ⊢ {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} ** ext1 ω ** case h Ω : Type u_1 β : Type u_2 ι : Type u_3 m : MeasurableSpace Ω inst✝³ : LinearOrder ι f : Filtration ι m τ π : Ω → ι inst✝² : TopologicalSpace ι inst✝¹ : OrderTopology ι inst✝ : FirstCountableTopology ι hτ : IsStoppingTime f τ i : ι ω : Ω ⊢ ω ∈ {ω | τ ω < i} ↔ ω ∈ {ω | τ ω ≤ i} \ {ω | τ ω = i} ** simp only [lt_iff_le_and_ne, Set.mem_setOf_eq, Set.mem_diff] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range' ** 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 h_countable : Set.Countable (Set.range τ) i : ι ⊢ MeasurableSet {ω | τ ω = i} ** exact hτ.measurableSet_eq_of_countable_range h_countable i ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range' ** 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 h_countable : Set.Countable (Set.range τ) i : ι ⊢ MeasurableSet {ω | i ≤ τ ω} ** have : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω} := by
ext1 ω
simp only [le_iff_lt_or_eq, Set.mem_setOf_eq, Set.mem_union]
rw [@eq_comm _ i, or_comm] ** 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 h_countable : Set.Countable (Set.range τ) i : ι this : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω} ⊢ MeasurableSet {ω | i ≤ τ ω} ** rw [this] ** 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 h_countable : Set.Countable (Set.range τ) i : ι this : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω} ⊢ MeasurableSet ({ω | τ ω = i} ∪ {ω | i < τ ω}) ** exact (hτ.measurableSet_eq_of_countable_range' h_countable i).union (hτ.measurableSet_gt' 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 h_countable : Set.Countable (Set.range τ) i : ι ⊢ {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω} ** ext1 ω ** case h 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 h_countable : Set.Countable (Set.range τ) i : ι ω : Ω ⊢ ω ∈ {ω | i ≤ τ ω} ↔ ω ∈ {ω | τ ω = i} ∪ {ω | i < τ ω} ** simp only [le_iff_lt_or_eq, Set.mem_setOf_eq, Set.mem_union] ** case h 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 h_countable : Set.Countable (Set.range τ) i : ι ω : Ω ⊢ i < τ ω ∨ i = τ ω ↔ τ ω = i ∨ i < τ ω ** rw [@eq_comm _ i, or_comm] ** Qed
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MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range' ** 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 h_countable : Set.Countable (Set.range τ) i : ι ⊢ MeasurableSet {ω | τ ω < i} ** have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} := by
ext1 ω
simp only [lt_iff_le_and_ne, Set.mem_setOf_eq, Set.mem_diff] ** 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 h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} ⊢ MeasurableSet {ω | τ ω < i} ** rw [this] ** 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 h_countable : Set.Countable (Set.range τ) i : ι this : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} ⊢ MeasurableSet ({ω | τ ω ≤ i} \ {ω | τ ω = i}) ** exact (hτ.measurableSet_le' i).diff (hτ.measurableSet_eq_of_countable_range' h_countable 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 h_countable : Set.Countable (Set.range τ) i : ι ⊢ {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i} ** ext1 ω ** case h 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 h_countable : Set.Countable (Set.range τ) i : ι ω : Ω ⊢ ω ∈ {ω | τ ω < i} ↔ ω ∈ {ω | τ ω ≤ i} \ {ω | τ ω = i} ** simp only [lt_iff_le_and_ne, Set.mem_setOf_eq, Set.mem_diff] ** Qed
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MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable_range ** 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 h_countable : Set.Countable (Set.range τ) ⊢ IsStoppingTime.measurableSpace hτ ≤ m ** intro s hs ** 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : MeasurableSet s ⊢ MeasurableSet s ** change ∀ i, MeasurableSet[f i] (s ∩ {ω | τ ω ≤ i}) at hs ** 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ⊢ MeasurableSet s ** rw [(_ : s = ⋃ i ∈ Set.range τ, s ∩ {ω | τ ω ≤ 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ⊢ MeasurableSet (⋃ i ∈ Set.range τ, s ∩ {ω | τ ω ≤ i}) ** exact MeasurableSet.biUnion h_countable fun i _ => f.le i _ (hs 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ⊢ s = ⋃ i ∈ Set.range τ, s ∩ {ω | τ ω ≤ i} ** ext ω ** case h 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω ⊢ ω ∈ s ↔ ω ∈ ⋃ i ∈ Set.range τ, s ∩ {ω | τ ω ≤ i} ** constructor <;> rw [Set.mem_iUnion] ** case h.mp 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω ⊢ ω ∈ s → ∃ i, ω ∈ ⋃ (_ : i ∈ Set.range τ), s ∩ {ω | τ ω ≤ i} ** exact fun hx => ⟨τ ω, by simpa using hx⟩ ** 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω hx : ω ∈ s ⊢ ω ∈ ⋃ (_ : τ ω ∈ Set.range τ), s ∩ {ω_1 | τ ω_1 ≤ τ ω} ** simpa using hx ** case h.mpr 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω ⊢ (∃ i, ω ∈ ⋃ (_ : i ∈ Set.range τ), s ∩ {ω | τ ω ≤ i}) → ω ∈ s ** rintro ⟨i, hx⟩ ** case h.mpr.intro 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω i : ι hx : ω ∈ ⋃ (_ : i ∈ Set.range τ), s ∩ {ω | τ ω ≤ i} ⊢ ω ∈ s ** simp only [Set.mem_range, Set.iUnion_exists, Set.mem_iUnion, Set.mem_inter_iff,
Set.mem_setOf_eq, exists_prop, exists_and_right] at hx ** case h.mpr.intro 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 h_countable : Set.Countable (Set.range τ) s : Set Ω hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i}) ω : Ω i : ι hx : (∃ y, τ y = i) ∧ ω ∈ s ∧ τ ω ≤ i ⊢ ω ∈ s ** exact hx.2.1 ** Qed
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MeasureTheory.IsStoppingTime.measurableSpace_min ** 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 ⊢ IsStoppingTime.measurableSpace (_ : IsStoppingTime f fun ω => min (τ ω) (π ω)) = IsStoppingTime.measurableSpace hτ ⊓ IsStoppingTime.measurableSpace hπ ** refine' le_antisymm _ _ ** case refine'_1 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 ⊢ IsStoppingTime.measurableSpace (_ : IsStoppingTime f fun ω => min (τ ω) (π ω)) ≤ IsStoppingTime.measurableSpace hτ ⊓ IsStoppingTime.measurableSpace hπ ** exact le_inf (measurableSpace_mono _ hτ fun _ => min_le_left _ _)
(measurableSpace_mono _ hπ fun _ => min_le_right _ _) ** case refine'_2 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 ⊢ IsStoppingTime.measurableSpace hτ ⊓ IsStoppingTime.measurableSpace hπ ≤ IsStoppingTime.measurableSpace (_ : IsStoppingTime f fun ω => min (τ ω) (π ω)) ** intro s ** case refine'_2 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 ** change MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[hπ.measurableSpace] s →
MeasurableSet[(hτ.min hπ).measurableSpace] s ** case refine'_2 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 ** simp_rw [IsStoppingTime.measurableSet] ** case refine'_2 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 ⊢ ((∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})) ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | π ω ≤ i})) → ∀ (i : ι), MeasurableSet (s ∩ {ω | min (τ ω) (π ω) ≤ i}) ** have : ∀ i, {ω | min (τ ω) (π ω) ≤ i} = {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i} := by
intro i; ext1 ω; simp ** case refine'_2 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 this : ∀ (i : ι), {ω | min (τ ω) (π ω) ≤ i} = {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i} ⊢ ((∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})) ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | π ω ≤ i})) → ∀ (i : ι), MeasurableSet (s ∩ {ω | min (τ ω) (π ω) ≤ i}) ** simp_rw [this, Set.inter_union_distrib_left] ** case refine'_2 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 this : ∀ (i : ι), {ω | min (τ ω) (π ω) ≤ i} = {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i} ⊢ ((∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})) ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | π ω ≤ i})) → ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i} ∪ s ∩ {ω | π ω ≤ i}) ** exact fun h i => (h.left i).union (h.right 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 hπ : IsStoppingTime f π s : Set Ω ⊢ ∀ (i : ι), {ω | min (τ ω) (π ω) ≤ i} = {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i} ** intro 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 hπ : IsStoppingTime f π s : Set Ω i : ι ⊢ {ω | min (τ ω) (π ω) ≤ i} = {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i} ** ext1 ω ** case h 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 i : ι ω : Ω ⊢ ω ∈ {ω | min (τ ω) (π ω) ≤ i} ↔ ω ∈ {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i} ** simp ** Qed
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