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MeasureTheory.inducedOuterMeasure_eq_iInf ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α ⊢ ↑(inducedOuterMeasure m P0 m0) s = ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ** apply le_antisymm ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α ⊢ ↑(inducedOuterMeasure m P0 m0) s ≤ ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ** simp only [le_iInf_iff] ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α ⊢ ∀ (i : Set α) (i_1 : P i), s ⊆ i → ↑(inducedOuterMeasure m P0 m0) s ≤ m i i_1 ** intro t ht hs ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s t : Set α ht : P t hs : s ⊆ t ⊢ ↑(inducedOuterMeasure m P0 m0) s ≤ m t ht ** refine' le_trans (mono' _ hs) _ ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s t : Set α ht : P t hs : s ⊆ t ⊢ ↑(inducedOuterMeasure m P0 m0) t ≤ m t ht ** exact le_of_eq (inducedOuterMeasure_eq' _ msU m_mono _) ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α ⊢ ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ≤ ↑(inducedOuterMeasure m P0 m0) s ** refine' le_iInf _ ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α ⊢ ∀ (i : ℕ → Set α), ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ≤ ⨅ (_ : s ⊆ ⋃ i_1, i i_1), ∑' (i_1 : ℕ), extend m (i i_1) ** intro f ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α f : ℕ → Set α ⊢ ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ≤ ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), extend m (f i) ** refine' le_iInf _ ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α f : ℕ → Set α ⊢ s ⊆ ⋃ i, f i → ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ≤ ∑' (i : ℕ), extend m (f i) ** intro hf ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α f : ℕ → Set α hf : s ⊆ ⋃ i, f i ⊢ ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ≤ ∑' (i : ℕ), extend m (f i) ** refine' le_trans _ (extend_iUnion_le_tsum_nat' _ msU _) ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α f : ℕ → Set α hf : s ⊆ ⋃ i, f i ⊢ ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ≤ extend m (⋃ i, f i) ** refine' le_iInf _ ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α f : ℕ → Set α hf : s ⊆ ⋃ i, f i ⊢ ∀ (i : (fun s => P s) (⋃ i, f i)), ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ≤ m (⋃ i, f i) i ** intro h2f ** case a α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α f : ℕ → Set α hf : s ⊆ ⋃ i, f i h2f : (fun s => P s) (⋃ i, f i) ⊢ ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ≤ m (⋃ i, f i) h2f ** refine' iInf_le_of_le _ (iInf_le_of_le h2f <| iInf_le _ hf) ** Qed
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MeasureTheory.inducedOuterMeasure_preimage ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ f : α ≃ α Pm : ∀ (s : Set α), P (↑f ⁻¹' s) ↔ P s mm : ∀ (s : Set α) (hs : P s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs A : Set α ⊢ ↑(inducedOuterMeasure m P0 m0) (↑f ⁻¹' A) = ↑(inducedOuterMeasure m P0 m0) A ** rw [inducedOuterMeasure_eq_iInf _ msU m_mono, inducedOuterMeasure_eq_iInf _ msU m_mono] ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ f : α ≃ α Pm : ∀ (s : Set α), P (↑f ⁻¹' s) ↔ P s mm : ∀ (s : Set α) (hs : P s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs A : Set α ⊢ ⨅ t, ⨅ (ht : P t), ⨅ (_ : ↑f ⁻¹' A ⊆ t), m t ht = ⨅ t, ⨅ (ht : P t), ⨅ (_ : A ⊆ t), m t ht ** symm ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ f : α ≃ α Pm : ∀ (s : Set α), P (↑f ⁻¹' s) ↔ P s mm : ∀ (s : Set α) (hs : P s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs A : Set α ⊢ ⨅ t, ⨅ (ht : P t), ⨅ (_ : A ⊆ t), m t ht = ⨅ t, ⨅ (ht : P t), ⨅ (_ : ↑f ⁻¹' A ⊆ t), m t ht ** refine' f.injective.preimage_surjective.iInf_congr (preimage f) fun s => _ ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ f : α ≃ α Pm : ∀ (s : Set α), P (↑f ⁻¹' s) ↔ P s mm : ∀ (s : Set α) (hs : P s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs A s : Set α ⊢ ⨅ (ht : P (↑f ⁻¹' s)), ⨅ (_ : ↑f ⁻¹' A ⊆ ↑f ⁻¹' s), m (↑f ⁻¹' s) ht = ⨅ (ht : P s), ⨅ (_ : A ⊆ s), m s ht ** refine' iInf_congr_Prop (Pm s) _ ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ f : α ≃ α Pm : ∀ (s : Set α), P (↑f ⁻¹' s) ↔ P s mm : ∀ (s : Set α) (hs : P s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs A s : Set α ⊢ ∀ (x : P s), ⨅ (_ : ↑f ⁻¹' A ⊆ ↑f ⁻¹' s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = ⨅ (_ : A ⊆ s), m s x ** intro hs ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ f : α ≃ α Pm : ∀ (s : Set α), P (↑f ⁻¹' s) ↔ P s mm : ∀ (s : Set α) (hs : P s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs A s : Set α hs : P s ⊢ ⨅ (_ : ↑f ⁻¹' A ⊆ ↑f ⁻¹' s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = ⨅ (_ : A ⊆ s), m s hs ** refine' iInf_congr_Prop f.surjective.preimage_subset_preimage_iff _ ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ f : α ≃ α Pm : ∀ (s : Set α), P (↑f ⁻¹' s) ↔ P s mm : ∀ (s : Set α) (hs : P s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs A s : Set α hs : P s ⊢ A ⊆ s → m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs ** intro _ ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ f : α ≃ α Pm : ∀ (s : Set α), P (↑f ⁻¹' s) ↔ P s mm : ∀ (s : Set α) (hs : P s), m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs A s : Set α hs : P s x✝ : A ⊆ s ⊢ m (↑f ⁻¹' s) (_ : P (↑f ⁻¹' s)) = m s hs ** exact mm s hs ** Qed
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MeasureTheory.inducedOuterMeasure_exists_set ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 ⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε ** have h := ENNReal.lt_add_right hs hε ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 h : ↑(inducedOuterMeasure m P0 m0) s < ↑(inducedOuterMeasure m P0 m0) s + ε ⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε ** conv at h =>
lhs
rw [inducedOuterMeasure_eq_iInf _ msU m_mono] ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 h : ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht < ↑(inducedOuterMeasure m P0 m0) s + ε ⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε ** simp only [iInf_lt_iff] at h ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 h : ∃ i h i_1, m i (_ : P i) < ↑(inducedOuterMeasure m P0 m0) s + ε ⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε ** rcases h with ⟨t, h1t, h2t, h3t⟩ ** case intro.intro.intro α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤ ε : ℝ≥0∞ hε : ε ≠ 0 t : Set α h1t : P t h2t : s ⊆ t h3t : m t (_ : P t) < ↑(inducedOuterMeasure m P0 m0) s + ε ⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε ** exact
⟨t, h1t, h2t, le_trans (le_of_eq <| inducedOuterMeasure_eq' _ msU m_mono h1t) (le_of_lt h3t)⟩ ** Qed
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MeasureTheory.inducedOuterMeasure_caratheodory ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α ⊢ MeasurableSet s ↔ ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t ** rw [isCaratheodory_iff_le] ** α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α ⊢ (∀ (t : Set α), ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t) ↔ ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t ** constructor ** case mp α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α ⊢ (∀ (t : Set α), ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t) → ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t ** intro h t _ht ** case mp α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t t : Set α _ht : P t ⊢ ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t ** exact h t ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α ⊢ (∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t) → ∀ (t : Set α), ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t ** intro h u ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t u : Set α ⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \ s) ≤ ↑(inducedOuterMeasure m P0 m0) u ** conv_rhs => rw [inducedOuterMeasure_eq_iInf _ msU m_mono] ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t u : Set α ⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \ s) ≤ ⨅ t, ⨅ (ht : P t), ⨅ (_ : u ⊆ t), m t ht ** refine' le_iInf _ ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t u : Set α ⊢ ∀ (i : Set α), ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \ s) ≤ ⨅ (ht : P i), ⨅ (_ : u ⊆ i), m i ht ** intro t ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t u t : Set α ⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \ s) ≤ ⨅ (ht : P t), ⨅ (_ : u ⊆ t), m t ht ** refine' le_iInf _ ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t u t : Set α ⊢ ∀ (i : P t), ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \ s) ≤ ⨅ (_ : u ⊆ t), m t i ** intro ht ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t u t : Set α ht : P t ⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \ s) ≤ ⨅ (_ : u ⊆ t), m t ht ** refine' le_iInf _ ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t u t : Set α ht : P t ⊢ u ⊆ t → ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \ s) ≤ m t ht ** intro h2t ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t u t : Set α ht : P t h2t : u ⊆ t ⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \ s) ≤ m t ht ** refine' le_trans _ (le_trans (h t ht) <| le_of_eq <| inducedOuterMeasure_eq' _ msU m_mono ht) ** case mpr α : Type u_1 P : Set α → Prop m : (s : Set α) → P s → ℝ≥0∞ P0 : P ∅ m0 : m ∅ P0 = 0 PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i) mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : P (⋃ i, f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i)) msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) (_ : P (⋃ i, f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i)) m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂ s : Set α h : ∀ (t : Set α), P t → ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ≤ ↑(inducedOuterMeasure m P0 m0) t u t : Set α ht : P t h2t : u ⊆ t ⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \ s) ≤ ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \ s) ** refine'
add_le_add (mono' _ <| Set.inter_subset_inter_left _ h2t)
(mono' _ <| diff_subset_diff_left h2t) ** Qed
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MeasureTheory.extend_iUnion_le_tsum_nat ** α : Type u_1 inst✝ : MeasurableSpace α m : (s : Set α) → MeasurableSet s → ℝ≥0∞ m0 : m ∅ (_ : MeasurableSet ∅) = 0 mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : MeasurableSet (⋃ b, f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i)) ⊢ ∀ (s : ℕ → Set α), extend m (⋃ i, s i) ≤ ∑' (i : ℕ), extend m (s i) ** refine' extend_iUnion_le_tsum_nat' MeasurableSet.iUnion _ ** α : Type u_1 inst✝ : MeasurableSpace α m : (s : Set α) → MeasurableSet s → ℝ≥0∞ m0 : m ∅ (_ : MeasurableSet ∅) = 0 mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : MeasurableSet (⋃ b, f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i)) ⊢ ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), m (⋃ i, f i) (_ : MeasurableSet (⋃ b, f b)) ≤ ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i)) ** intro f h ** α : Type u_1 inst✝ : MeasurableSpace α m : (s : Set α) → MeasurableSet s → ℝ≥0∞ m0 : m ∅ (_ : MeasurableSet ∅) = 0 mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : MeasurableSet (⋃ b, f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i)) f : ℕ → Set α h : ∀ (i : ℕ), MeasurableSet (f i) ⊢ m (⋃ i, f i) (_ : MeasurableSet (⋃ b, f b)) ≤ ∑' (i : ℕ), m (f i) (_ : ?m.314738 (f i)) ** simp (config := { singlePass := true }) [iUnion_disjointed.symm] ** α : Type u_1 inst✝ : MeasurableSpace α m : (s : Set α) → MeasurableSet s → ℝ≥0∞ m0 : m ∅ (_ : MeasurableSet ∅) = 0 mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : MeasurableSet (⋃ b, f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i)) f : ℕ → Set α h : ∀ (i : ℕ), MeasurableSet (f i) ⊢ m (⋃ n, disjointed (fun n => f n) n) (_ : MeasurableSet (⋃ n, disjointed (fun n => f n) n)) ≤ ∑' (i : ℕ), m (f i) (_ : ?m.314738 (f i)) ** rw [mU (MeasurableSet.disjointed h) (disjoint_disjointed _)] ** α : Type u_1 inst✝ : MeasurableSpace α m : (s : Set α) → MeasurableSet s → ℝ≥0∞ m0 : m ∅ (_ : MeasurableSet ∅) = 0 mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : MeasurableSet (⋃ b, f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i)) f : ℕ → Set α h : ∀ (i : ℕ), MeasurableSet (f i) ⊢ ∑' (i : ℕ), m (disjointed (fun i => f i) i) (_ : MeasurableSet (disjointed (fun i => f i) i)) ≤ ∑' (i : ℕ), m (f i) (_ : ?m.314738 (f i)) ** refine' ENNReal.tsum_le_tsum fun i => _ ** α : Type u_1 inst✝ : MeasurableSpace α m : (s : Set α) → MeasurableSet s → ℝ≥0∞ m0 : m ∅ (_ : MeasurableSet ∅) = 0 mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : MeasurableSet (⋃ b, f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i)) f : ℕ → Set α h : ∀ (i : ℕ), MeasurableSet (f i) i : ℕ ⊢ m (disjointed (fun i => f i) i) (_ : MeasurableSet (disjointed (fun i => f i) i)) ≤ m (f i) (_ : ?m.314738 (f i)) ** rw [← extend_eq m, ← extend_eq m] ** α : Type u_1 inst✝ : MeasurableSpace α m : (s : Set α) → MeasurableSet s → ℝ≥0∞ m0 : m ∅ (_ : MeasurableSet ∅) = 0 mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) (_ : MeasurableSet (⋃ b, f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i)) f : ℕ → Set α h : ∀ (i : ℕ), MeasurableSet (f i) i : ℕ ⊢ extend m (disjointed (fun i => f i) i) ≤ extend m (f i) ** exact extend_mono m0 mU (MeasurableSet.disjointed h _) (disjointed_le f _) ** Qed
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MeasureTheory.OuterMeasure.le_trim ** α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α ⊢ m ≤ trim m ** apply le_ofFunction.mpr ** α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α ⊢ ∀ (s : Set α), ↑m s ≤ extend (fun s x => ↑m s) s α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α ⊢ extend (fun s x => ↑m s) ∅ = 0 ** intro s ** α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α s : Set α ⊢ ↑m s ≤ extend (fun s x => ↑m s) s α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α ⊢ extend (fun s x => ↑m s) ∅ = 0 ** apply le_iInf ** case h α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α s : Set α ⊢ ∀ (i : (fun s => MeasurableSet s) s), ↑m s ≤ (fun s x => ↑m s) s i α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α ⊢ extend (fun s x => ↑m s) ∅ = 0 ** simp only [le_refl, implies_true] ** α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α ⊢ extend (fun s x => ↑m s) ∅ = 0 ** apply extend_empty <;> simp ** Qed
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MeasureTheory.OuterMeasure.trim_congr ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α H : ∀ {s : Set α}, MeasurableSet s → ↑m₁ s = ↑m₂ s ⊢ trim m₁ = trim m₂ ** unfold trim ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α H : ∀ {s : Set α}, MeasurableSet s → ↑m₁ s = ↑m₂ s ⊢ inducedOuterMeasure (fun s x => ↑m₁ s) (_ : MeasurableSet ∅) (_ : ↑m₁ ∅ = 0) = inducedOuterMeasure (fun s x => ↑m₂ s) (_ : MeasurableSet ∅) (_ : ↑m₂ ∅ = 0) ** congr ** case e_m α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α H : ∀ {s : Set α}, MeasurableSet s → ↑m₁ s = ↑m₂ s ⊢ (fun s x => ↑m₁ s) = fun s x => ↑m₂ s ** funext s hs ** case e_m.h.h α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α H : ∀ {s : Set α}, MeasurableSet s → ↑m₁ s = ↑m₂ s s : Set α hs : MeasurableSet s ⊢ ↑m₁ s = ↑m₂ s ** exact H hs ** Qed
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MeasureTheory.OuterMeasure.le_trim_iff ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α ⊢ m₁ ≤ trim m₂ ↔ ∀ (s : Set α), MeasurableSet s → ↑m₁ s ≤ ↑m₂ s ** let me := extend (fun s (_p : MeasurableSet s) => measureOf m₂ s) ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s ⊢ m₁ ≤ trim m₂ ↔ ∀ (s : Set α), MeasurableSet s → ↑m₁ s ≤ ↑m₂ s ** have me_empty : me ∅ = 0 := by apply extend_empty; simp; simp ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s me_empty : me ∅ = 0 ⊢ m₁ ≤ trim m₂ ↔ ∀ (s : Set α), MeasurableSet s → ↑m₁ s ≤ ↑m₂ s ** have : m₁ ≤ OuterMeasure.ofFunction me me_empty ↔
(∀ (s : Set α), measureOf m₁ s ≤ me s) := le_ofFunction ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s me_empty : me ∅ = 0 this : m₁ ≤ OuterMeasure.ofFunction me me_empty ↔ ∀ (s : Set α), ↑m₁ s ≤ me s ⊢ m₁ ≤ trim m₂ ↔ ∀ (s : Set α), MeasurableSet s → ↑m₁ s ≤ ↑m₂ s ** apply this.trans ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s me_empty : me ∅ = 0 this : m₁ ≤ OuterMeasure.ofFunction me me_empty ↔ ∀ (s : Set α), ↑m₁ s ≤ me s ⊢ (∀ (s : Set α), ↑m₁ s ≤ me s) ↔ ∀ (s : Set α), MeasurableSet s → ↑m₁ s ≤ ↑m₂ s ** apply forall_congr' ** case h α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s me_empty : me ∅ = 0 this : m₁ ≤ OuterMeasure.ofFunction me me_empty ↔ ∀ (s : Set α), ↑m₁ s ≤ me s ⊢ ∀ (a : Set α), ↑m₁ a ≤ me a ↔ MeasurableSet a → ↑m₁ a ≤ ↑m₂ a ** intro s ** case h α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s me_empty : me ∅ = 0 this : m₁ ≤ OuterMeasure.ofFunction me me_empty ↔ ∀ (s : Set α), ↑m₁ s ≤ me s s : Set α ⊢ ↑m₁ s ≤ me s ↔ MeasurableSet s → ↑m₁ s ≤ ↑m₂ s ** apply le_iInf_iff ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s ⊢ me ∅ = 0 ** apply extend_empty ** case m0 α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s ⊢ ↑m₂ ∅ = 0 case P0 α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s ⊢ MeasurableSet ∅ ** simp ** case P0 α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α me : Set α → ℝ≥0∞ := extend fun s _p => ↑m₂ s ⊢ MeasurableSet ∅ ** simp ** Qed
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MeasureTheory.OuterMeasure.trim_le_trim_iff ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ : OuterMeasure α s : Set α hs : MeasurableSet s ⊢ ↑(trim m₁) s ≤ ↑m₂ s ↔ ↑m₁ s ≤ ↑m₂ s ** rw [trim_eq _ hs] ** Qed
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MeasureTheory.OuterMeasure.trim_eq_iInf ** α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α s : Set α ⊢ ↑(trim m) s = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t ** simp (config := { singlePass := true }) only [iInf_comm] ** α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α s : Set α ⊢ ↑(trim m) s = ⨅ t, ⨅ (_ : MeasurableSet t), ⨅ (_ : s ⊆ t), ↑m t ** exact
inducedOuterMeasure_eq_iInf MeasurableSet.iUnion (fun f _ => m.iUnion_nat f)
(fun _ _ _ _ h => m.mono h) s ** Qed
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MeasureTheory.OuterMeasure.trim_eq_iInf' ** α : Type u_1 inst✝ : MeasurableSpace α m : OuterMeasure α s : Set α ⊢ ↑(trim m) s = ⨅ t, ↑m ↑t ** simp [iInf_subtype, iInf_and, trim_eq_iInf] ** Qed
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MeasureTheory.OuterMeasure.trim_sum_ge ** α : Type u_1 inst✝ : MeasurableSpace α m✝ : OuterMeasure α ι : Type u_2 m : ι → OuterMeasure α s : Set α ⊢ ↑(sum fun i => trim (m i)) s ≤ ↑(trim (sum m)) s ** simp [trim_eq_iInf] ** α : Type u_1 inst✝ : MeasurableSpace α m✝ : OuterMeasure α ι : Type u_2 m : ι → OuterMeasure α s : Set α ⊢ ∀ (i : Set α), s ⊆ i → MeasurableSet i → ∑' (i : ι), ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑(m i) t ≤ ∑' (i_3 : ι), ↑(m i_3) i ** exact fun t st ht =>
ENNReal.tsum_le_tsum fun i => iInf_le_of_le t <| iInf_le_of_le st <| iInf_le _ ht ** Qed
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MeasureTheory.OuterMeasure.exists_measurable_superset_eq_trim ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ↑(trim m) s ** simp only [trim_eq_iInf] ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t ** set ms := ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), m t ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms ** by_cases hs : ms = ∞ ** case pos α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ms = ⊤ ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms ** simp only [hs] ** case pos α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ms = ⊤ ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ⊤ ** simp only [iInf_eq_top] at hs ** case pos α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ∀ (i : Set α), s ⊆ i → MeasurableSet i → ↑m i = ⊤ ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ⊤ ** exact ⟨univ, subset_univ s, MeasurableSet.univ, hs _ (subset_univ s) MeasurableSet.univ⟩ ** case neg α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms ** have : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r := by
intro r hs
have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs
rcases this with ⟨t, hmt, hin, hlt⟩
exists t ** case neg α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms ** have : ∀ n : ℕ, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < ms + (n : ℝ≥0∞)⁻¹ := by
intro n
refine' this _ (ENNReal.lt_add_right hs _)
simp ** case neg α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r this : ∀ (n : ℕ), ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹ ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms ** choose t hsub hm hm' using this ** case neg α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r t : ℕ → Set α hsub : ∀ (n : ℕ), s ⊆ t n hm : ∀ (n : ℕ), MeasurableSet (t n) hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹ ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = ms ** refine' ⟨⋂ n, t n, subset_iInter hsub, MeasurableSet.iInter hm, _⟩ ** case neg α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r t : ℕ → Set α hsub : ∀ (n : ℕ), s ⊆ t n hm : ∀ (n : ℕ), MeasurableSet (t n) hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹ ⊢ ↑m (⋂ n, t n) = ms ** have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=
tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero ** case neg α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r t : ℕ → Set α hsub : ∀ (n : ℕ), s ⊆ t n hm : ∀ (n : ℕ), MeasurableSet (t n) hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹ this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 (ms + 0)) ⊢ ↑m (⋂ n, t n) = ms ** rw [add_zero] at this ** case neg α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r t : ℕ → Set α hsub : ∀ (n : ℕ), s ⊆ t n hm : ∀ (n : ℕ), MeasurableSet (t n) hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹ this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms) ⊢ ↑m (⋂ n, t n) = ms ** refine' le_antisymm (ge_of_tendsto' this fun n => _) _ ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ ⊢ ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r ** intro r hs ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs✝ : ¬ms = ⊤ r : ℝ≥0∞ hs : r > ms ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r ** have : ∃t, MeasurableSet t ∧ s ⊆ t ∧ measureOf m t < r := by simpa [iInf_lt_iff] using hs ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs✝ : ¬ms = ⊤ r : ℝ≥0∞ hs : r > ms this : ∃ t, MeasurableSet t ∧ s ⊆ t ∧ ↑m t < r ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r ** rcases this with ⟨t, hmt, hin, hlt⟩ ** case intro.intro.intro α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs✝ : ¬ms = ⊤ r : ℝ≥0∞ hs : r > ms t : Set α hmt : MeasurableSet t hin : s ⊆ t hlt : ↑m t < r ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r ** exists t ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs✝ : ¬ms = ⊤ r : ℝ≥0∞ hs : r > ms ⊢ ∃ t, MeasurableSet t ∧ s ⊆ t ∧ ↑m t < r ** simpa [iInf_lt_iff] using hs ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r ⊢ ∀ (n : ℕ), ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹ ** intro n ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r n : ℕ ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < ms + (↑n)⁻¹ ** refine' this _ (ENNReal.lt_add_right hs _) ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r n : ℕ ⊢ (↑n)⁻¹ ≠ 0 ** simp ** case neg.refine'_1 α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r t : ℕ → Set α hsub : ∀ (n : ℕ), s ⊆ t n hm : ∀ (n : ℕ), MeasurableSet (t n) hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹ this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms) n : ℕ ⊢ ↑m (⋂ n, t n) ≤ ms + (↑n)⁻¹ ** exact le_trans (m.mono' <| iInter_subset t n) (hm' n).le ** case neg.refine'_2 α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r t : ℕ → Set α hsub : ∀ (n : ℕ), s ⊆ t n hm : ∀ (n : ℕ), MeasurableSet (t n) hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹ this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms) ⊢ ms ≤ ↑m (⋂ n, t n) ** refine' iInf_le_of_le (⋂ n, t n) _ ** case neg.refine'_2 α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r t : ℕ → Set α hsub : ∀ (n : ℕ), s ⊆ t n hm : ∀ (n : ℕ), MeasurableSet (t n) hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹ this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms) ⊢ ⨅ (_ : s ⊆ ⋂ n, t n), ⨅ (_ : MeasurableSet (⋂ n, t n)), ↑m (⋂ n, t n) ≤ ↑m (⋂ n, t n) ** refine' iInf_le_of_le (subset_iInter hsub) _ ** case neg.refine'_2 α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑m t hs : ¬ms = ⊤ this✝ : ∀ (r : ℝ≥0∞), r > ms → ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t < r t : ℕ → Set α hsub : ∀ (n : ℕ), s ⊆ t n hm : ∀ (n : ℕ), MeasurableSet (t n) hm' : ∀ (n : ℕ), ↑m (t n) < ms + (↑n)⁻¹ this : Tendsto (fun n => ms + (↑n)⁻¹) atTop (𝓝 ms) ⊢ ⨅ (_ : MeasurableSet (⋂ n, t n)), ↑m (⋂ n, t n) ≤ ↑m (⋂ n, t n) ** refine' iInf_le _ (MeasurableSet.iInter hm) ** Qed
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MeasureTheory.OuterMeasure.exists_measurable_superset_of_trim_eq_zero ** α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α h : ↑(trim m) s = 0 ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = 0 ** rcases exists_measurable_superset_eq_trim m s with ⟨t, hst, ht, hm⟩ ** case intro.intro.intro α : Type u_1 inst✝ : MeasurableSpace α m✝ m : OuterMeasure α s : Set α h : ↑(trim m) s = 0 t : Set α hst : s ⊆ t ht : MeasurableSet t hm : ↑m t = ↑(trim m) s ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = 0 ** exact ⟨t, hst, ht, h ▸ hm⟩ ** Qed
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MeasureTheory.OuterMeasure.exists_measurable_superset_forall_eq_trim ** α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s : Set α ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s ** choose t hst ht hμt using fun i => (μ i).exists_measurable_superset_eq_trim s ** α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s : Set α t : ι → Set α hst : ∀ (i : ι), s ⊆ t i ht : ∀ (i : ι), MeasurableSet (t i) hμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s ** replace hst := subset_iInter hst ** α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s : Set α t : ι → Set α ht : ∀ (i : ι), MeasurableSet (t i) hμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s hst : s ⊆ ⋂ i, t i ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s ** replace ht := MeasurableSet.iInter ht ** α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s : Set α t : ι → Set α hμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s hst : s ⊆ ⋂ i, t i ht : MeasurableSet (⋂ b, t b) ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑(μ i) t = ↑(trim (μ i)) s ** refine' ⟨⋂ i, t i, hst, ht, fun i => le_antisymm _ _⟩ ** case refine'_1 α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s : Set α t : ι → Set α hμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s hst : s ⊆ ⋂ i, t i ht : MeasurableSet (⋂ b, t b) i : ι ⊢ ↑(μ i) (⋂ i, t i) ≤ ↑(trim (μ i)) s case refine'_2 α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s : Set α t : ι → Set α hμt : ∀ (i : ι), ↑(μ i) (t i) = ↑(trim (μ i)) s hst : s ⊆ ⋂ i, t i ht : MeasurableSet (⋂ b, t b) i : ι ⊢ ↑(trim (μ i)) s ≤ ↑(μ i) (⋂ i, t i) ** exacts [hμt i ▸ (μ i).mono (iInter_subset _ _), (mono' _ hst).trans_eq ((μ i).trim_eq ht)] ** Qed
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MeasureTheory.OuterMeasure.trim_binop ** α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ m₃ : OuterMeasure α op : ℝ≥0∞ → ℝ≥0∞ → ℝ≥0∞ h : ∀ (s : Set α), ↑m₁ s = op (↑m₂ s) (↑m₃ s) s : Set α ⊢ ↑(trim m₁) s = op (↑(trim m₂) s) (↑(trim m₃) s) ** rcases exists_measurable_superset_forall_eq_trim ![m₁, m₂, m₃] s with ⟨t, _hst, _ht, htm⟩ ** case intro.intro.intro α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ m₃ : OuterMeasure α op : ℝ≥0∞ → ℝ≥0∞ → ℝ≥0∞ h : ∀ (s : Set α), ↑m₁ s = op (↑m₂ s) (↑m₃ s) s t : Set α _hst : s ⊆ t _ht : MeasurableSet t htm : ∀ (i : Fin (Nat.succ (Nat.succ (Nat.succ 0)))), ↑(Matrix.vecCons m₁ ![m₂, m₃] i) t = ↑(trim (Matrix.vecCons m₁ ![m₂, m₃] i)) s ⊢ ↑(trim m₁) s = op (↑(trim m₂) s) (↑(trim m₃) s) ** simp only [Fin.forall_fin_succ, Matrix.cons_val_zero, Matrix.cons_val_succ] at htm ** case intro.intro.intro α : Type u_1 inst✝ : MeasurableSpace α m m₁ m₂ m₃ : OuterMeasure α op : ℝ≥0∞ → ℝ≥0∞ → ℝ≥0∞ h : ∀ (s : Set α), ↑m₁ s = op (↑m₂ s) (↑m₃ s) s t : Set α _hst : s ⊆ t _ht : MeasurableSet t htm : ↑m₁ t = ↑(trim m₁) s ∧ ↑m₂ t = ↑(trim m₂) s ∧ ↑m₃ t = ↑(trim m₃) s ∧ ∀ (i : Fin 0), ↑![] t = ↑(trim ![]) s ⊢ ↑(trim m₁) s = op (↑(trim m₂) s) (↑(trim m₃) s) ** rw [← htm.1, ← htm.2.1, ← htm.2.2.1, h] ** Qed
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MeasureTheory.OuterMeasure.trim_iSup ** α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α ⊢ trim (⨆ i, μ i) = ⨆ i, trim (μ i) ** simp_rw [← @iSup_plift_down _ ι] ** α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α ⊢ trim (⨆ i, μ i.down) = ⨆ i, trim (μ i.down) ** ext1 s ** case h α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s : Set α ⊢ ↑(trim (⨆ i, μ i.down)) s = ↑(⨆ i, trim (μ i.down)) s ** obtain ⟨t, _, _, hμt⟩ :=
exists_measurable_superset_forall_eq_trim
(Option.elim' (⨆ i, μ (PLift.down i)) (μ ∘ PLift.down)) s ** case h.intro.intro.intro α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s t : Set α left✝¹ : s ⊆ t left✝ : MeasurableSet t hμt : ∀ (i : Option (PLift ι)), ↑(Option.elim' (⨆ i, μ i.down) (μ ∘ PLift.down) i) t = ↑(trim (Option.elim' (⨆ i, μ i.down) (μ ∘ PLift.down) i)) s ⊢ ↑(trim (⨆ i, μ i.down)) s = ↑(⨆ i, trim (μ i.down)) s ** simp only [Option.forall, Option.elim'] at hμt ** case h.intro.intro.intro α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s t : Set α left✝¹ : s ⊆ t left✝ : MeasurableSet t hμt : ↑(⨆ i, μ i.down) t = ↑(trim (⨆ i, μ i.down)) s ∧ ∀ (x : PLift ι), ↑((μ ∘ PLift.down) x) t = ↑(trim ((μ ∘ PLift.down) x)) s ⊢ ↑(trim (⨆ i, μ i.down)) s = ↑(⨆ i, trim (μ i.down)) s ** simp only [iSup_apply, ← hμt.1] ** case h.intro.intro.intro α : Type u_1 inst✝¹ : MeasurableSpace α m : OuterMeasure α ι : Sort u_2 inst✝ : Countable ι μ : ι → OuterMeasure α s t : Set α left✝¹ : s ⊆ t left✝ : MeasurableSet t hμt : ↑(⨆ i, μ i.down) t = ↑(trim (⨆ i, μ i.down)) s ∧ ∀ (x : PLift ι), ↑((μ ∘ PLift.down) x) t = ↑(trim ((μ ∘ PLift.down) x)) s ⊢ ⨆ i, ↑(μ i.down) t = ⨆ i, ↑(trim (μ i.down)) s ** exact iSup_congr hμt.2 ** Qed
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MeasureTheory.OuterMeasure.restrict_trim ** α : Type u_1 inst✝ : MeasurableSpace α m μ : OuterMeasure α s : Set α hs : MeasurableSet s ⊢ trim (↑(restrict s) μ) = ↑(restrict s) (trim μ) ** refine' le_antisymm (fun t => _) (le_trim_iff.2 fun t ht => _) ** case refine'_1 α : Type u_1 inst✝ : MeasurableSpace α m μ : OuterMeasure α s : Set α hs : MeasurableSet s t : Set α ⊢ ↑(trim (↑(restrict s) μ)) t ≤ ↑(↑(restrict s) (trim μ)) t ** rw [restrict_apply] ** case refine'_1 α : Type u_1 inst✝ : MeasurableSpace α m μ : OuterMeasure α s : Set α hs : MeasurableSet s t : Set α ⊢ ↑(trim (↑(restrict s) μ)) t ≤ ↑(trim μ) (t ∩ s) ** rcases μ.exists_measurable_superset_eq_trim (t ∩ s) with ⟨t', htt', ht', hμt'⟩ ** case refine'_1.intro.intro.intro α : Type u_1 inst✝ : MeasurableSpace α m μ : OuterMeasure α s : Set α hs : MeasurableSet s t t' : Set α htt' : t ∩ s ⊆ t' ht' : MeasurableSet t' hμt' : ↑μ t' = ↑(trim μ) (t ∩ s) ⊢ ↑(trim (↑(restrict s) μ)) t ≤ ↑(trim μ) (t ∩ s) ** rw [← hμt'] ** case refine'_1.intro.intro.intro α : Type u_1 inst✝ : MeasurableSpace α m μ : OuterMeasure α s : Set α hs : MeasurableSet s t t' : Set α htt' : t ∩ s ⊆ t' ht' : MeasurableSet t' hμt' : ↑μ t' = ↑(trim μ) (t ∩ s) ⊢ ↑(trim (↑(restrict s) μ)) t ≤ ↑μ t' ** rw [inter_subset] at htt' ** case refine'_1.intro.intro.intro α : Type u_1 inst✝ : MeasurableSpace α m μ : OuterMeasure α s : Set α hs : MeasurableSet s t t' : Set α htt' : t ⊆ sᶜ ∪ t' ht' : MeasurableSet t' hμt' : ↑μ t' = ↑(trim μ) (t ∩ s) ⊢ ↑(trim (↑(restrict s) μ)) t ≤ ↑μ t' ** refine' (mono' _ htt').trans _ ** case refine'_1.intro.intro.intro α : Type u_1 inst✝ : MeasurableSpace α m μ : OuterMeasure α s : Set α hs : MeasurableSet s t t' : Set α htt' : t ⊆ sᶜ ∪ t' ht' : MeasurableSet t' hμt' : ↑μ t' = ↑(trim μ) (t ∩ s) ⊢ ↑(trim (↑(restrict s) μ)) (sᶜ ∪ t') ≤ ↑μ t' ** rw [trim_eq _ (hs.compl.union ht'), restrict_apply, union_inter_distrib_right, compl_inter_self,
Set.empty_union] ** case refine'_1.intro.intro.intro α : Type u_1 inst✝ : MeasurableSpace α m μ : OuterMeasure α s : Set α hs : MeasurableSet s t t' : Set α htt' : t ⊆ sᶜ ∪ t' ht' : MeasurableSet t' hμt' : ↑μ t' = ↑(trim μ) (t ∩ s) ⊢ ↑μ (t' ∩ s) ≤ ↑μ t' ** exact μ.mono' (inter_subset_left _ _) ** case refine'_2 α : Type u_1 inst✝ : MeasurableSpace α m μ : OuterMeasure α s : Set α hs : MeasurableSet s t : Set α ht : MeasurableSet t ⊢ ↑(↑(restrict s) (trim μ)) t ≤ ↑(↑(restrict s) μ) t ** rw [restrict_apply, trim_eq _ (ht.inter hs), restrict_apply] ** Qed
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Bundle.TotalSpace.mk_cast ** B : Type u_1 F : Type u_2 E : B → Type u_3 x x' : B h : x = x' b : E x ⊢ mk' F x' (cast (_ : E x = E x') b) = { proj := x, snd := b } ** subst h ** B : Type u_1 F : Type u_2 E : B → Type u_3 x : B b : E x ⊢ mk' F x (cast (_ : E x = E x) b) = { proj := x, snd := b } ** rfl ** Qed
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Bundle.TotalSpace.range_mk ** B : Type u_1 F : Type u_2 E : B → Type u_3 b : B ⊢ range (mk b) = proj ⁻¹' {b} ** apply Subset.antisymm ** case h₁ B : Type u_1 F : Type u_2 E : B → Type u_3 b : B ⊢ range (mk b) ⊆ proj ⁻¹' {b} ** rintro _ ⟨x, rfl⟩ ** case h₁.intro B : Type u_1 F : Type u_2 E : B → Type u_3 b : B x : E b ⊢ { proj := b, snd := x } ∈ proj ⁻¹' {b} ** rfl ** case h₂ B : Type u_1 F : Type u_2 E : B → Type u_3 b : B ⊢ proj ⁻¹' {b} ⊆ range (mk b) ** rintro ⟨_, x⟩ rfl ** case h₂.mk B : Type u_1 F : Type u_2 E : B → Type u_3 proj✝ : B x : E proj✝ ⊢ { proj := proj✝, snd := x } ∈ range (mk { proj := proj✝, snd := x }.proj) ** exact ⟨x, rfl⟩ ** Qed
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MeasureTheory.Measure.integral_comp_smul_of_nonneg ** E : Type u_1 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : MeasurableSpace E inst✝⁵ : BorelSpace E inst✝⁴ : FiniteDimensional ℝ E μ : Measure E inst✝³ : IsAddHaarMeasure μ F : Type u_2 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F s : Set E f : E → F R : ℝ hR : 0 ≤ R ⊢ ∫ (x : E), f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ ** rw [integral_comp_smul μ f R, abs_of_nonneg (inv_nonneg.2 (pow_nonneg hR _))] ** Qed
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MeasureTheory.Measure.integral_comp_inv_smul ** E : Type u_1 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : MeasurableSpace E inst✝⁵ : BorelSpace E inst✝⁴ : FiniteDimensional ℝ E μ : Measure E inst✝³ : IsAddHaarMeasure μ F : Type u_2 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F s : Set E f : E → F R : ℝ ⊢ ∫ (x : E), f (R⁻¹ • x) ∂μ = |R ^ finrank ℝ E| • ∫ (x : E), f x ∂μ ** rw [integral_comp_smul μ f R⁻¹, inv_pow, inv_inv] ** Qed
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MeasureTheory.Measure.integral_comp_inv_smul_of_nonneg ** E : Type u_1 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : MeasurableSpace E inst✝⁵ : BorelSpace E inst✝⁴ : FiniteDimensional ℝ E μ : Measure E inst✝³ : IsAddHaarMeasure μ F : Type u_2 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F s : Set E f : E → F R : ℝ hR : 0 ≤ R ⊢ ∫ (x : E), f (R⁻¹ • x) ∂μ = R ^ finrank ℝ E • ∫ (x : E), f x ∂μ ** rw [integral_comp_inv_smul μ f R, abs_of_nonneg (pow_nonneg hR _)] ** Qed
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MeasureTheory.Measure.integral_comp_mul_left ** E : Type u_1 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : MeasurableSpace E inst✝⁵ : BorelSpace E inst✝⁴ : FiniteDimensional ℝ E μ : Measure E inst✝³ : IsAddHaarMeasure μ F : Type u_2 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F s : Set E g : ℝ → F a : ℝ ⊢ ∫ (x : ℝ), g (a * x) = |a⁻¹| • ∫ (y : ℝ), g y ** simp_rw [← smul_eq_mul, Measure.integral_comp_smul, FiniteDimensional.finrank_self, pow_one] ** Qed
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MeasureTheory.Measure.integral_comp_inv_mul_left ** E : Type u_1 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : MeasurableSpace E inst✝⁵ : BorelSpace E inst✝⁴ : FiniteDimensional ℝ E μ : Measure E inst✝³ : IsAddHaarMeasure μ F : Type u_2 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F s : Set E g : ℝ → F a : ℝ ⊢ ∫ (x : ℝ), g (a⁻¹ * x) = |a| • ∫ (y : ℝ), g y ** simp_rw [← smul_eq_mul, Measure.integral_comp_inv_smul, FiniteDimensional.finrank_self, pow_one] ** Qed
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MeasureTheory.Measure.integral_comp_mul_right ** E : Type u_1 inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedSpace ℝ E inst✝⁶ : MeasurableSpace E inst✝⁵ : BorelSpace E inst✝⁴ : FiniteDimensional ℝ E μ : Measure E inst✝³ : IsAddHaarMeasure μ F : Type u_2 inst✝² : NormedAddCommGroup F inst✝¹ : NormedSpace ℝ F inst✝ : CompleteSpace F s : Set E g : ℝ → F a : ℝ ⊢ ∫ (x : ℝ), g (x * a) = |a⁻¹| • ∫ (y : ℝ), g y ** simpa only [mul_comm] using integral_comp_mul_left g a ** Qed
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MeasureTheory.integrable_comp_smul_iff ** F : Type u_1 inst✝⁶ : NormedAddCommGroup F E : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E inst✝¹ : FiniteDimensional ℝ E μ : Measure E inst✝ : IsAddHaarMeasure μ f : E → F R : ℝ hR : R ≠ 0 ⊢ (Integrable fun x => f (R • x)) ↔ Integrable f ** suffices
∀ {g : E → F} (hg : Integrable g μ) {S : ℝ} (hS : S ≠ 0), Integrable (fun x => g (S • x)) μ by
refine' ⟨fun hf => _, fun hf => this hf hR⟩
convert this hf (inv_ne_zero hR)
rw [← mul_smul, mul_inv_cancel hR, one_smul] ** F : Type u_1 inst✝⁶ : NormedAddCommGroup F E : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E inst✝¹ : FiniteDimensional ℝ E μ : Measure E inst✝ : IsAddHaarMeasure μ f : E → F R : ℝ hR : R ≠ 0 ⊢ ∀ {g : E → F}, Integrable g → ∀ {S : ℝ}, S ≠ 0 → Integrable fun x => g (S • x) ** intro g hg S hS ** F : Type u_1 inst✝⁶ : NormedAddCommGroup F E : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E inst✝¹ : FiniteDimensional ℝ E μ : Measure E inst✝ : IsAddHaarMeasure μ f : E → F R : ℝ hR : R ≠ 0 g : E → F hg : Integrable g S : ℝ hS : S ≠ 0 ⊢ Integrable fun x => g (S • x) ** let t := ((Homeomorph.smul (isUnit_iff_ne_zero.2 hS).unit).toMeasurableEquiv : E ≃ᵐ E) ** F : Type u_1 inst✝⁶ : NormedAddCommGroup F E : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E inst✝¹ : FiniteDimensional ℝ E μ : Measure E inst✝ : IsAddHaarMeasure μ f : E → F R : ℝ hR : R ≠ 0 g : E → F hg : Integrable g S : ℝ hS : S ≠ 0 t : E ≃ᵐ E := Homeomorph.toMeasurableEquiv (Homeomorph.smul (IsUnit.unit (_ : IsUnit S))) ⊢ Integrable g ** rwa [map_addHaar_smul μ hS, integrable_smul_measure _ ENNReal.ofReal_ne_top] ** F : Type u_1 inst✝⁶ : NormedAddCommGroup F E : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E inst✝¹ : FiniteDimensional ℝ E μ : Measure E inst✝ : IsAddHaarMeasure μ f : E → F R : ℝ hR : R ≠ 0 g : E → F hg : Integrable g S : ℝ hS : S ≠ 0 t : E ≃ᵐ E := Homeomorph.toMeasurableEquiv (Homeomorph.smul (IsUnit.unit (_ : IsUnit S))) ⊢ ENNReal.ofReal |(S ^ finrank ℝ E)⁻¹| ≠ 0 ** simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le, abs_pos] using inv_ne_zero (pow_ne_zero _ hS) ** F : Type u_1 inst✝⁶ : NormedAddCommGroup F E : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E inst✝¹ : FiniteDimensional ℝ E μ : Measure E inst✝ : IsAddHaarMeasure μ f : E → F R : ℝ hR : R ≠ 0 this : ∀ {g : E → F}, Integrable g → ∀ {S : ℝ}, S ≠ 0 → Integrable fun x => g (S • x) ⊢ (Integrable fun x => f (R • x)) ↔ Integrable f ** refine' ⟨fun hf => _, fun hf => this hf hR⟩ ** F : Type u_1 inst✝⁶ : NormedAddCommGroup F E : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E inst✝¹ : FiniteDimensional ℝ E μ : Measure E inst✝ : IsAddHaarMeasure μ f : E → F R : ℝ hR : R ≠ 0 this : ∀ {g : E → F}, Integrable g → ∀ {S : ℝ}, S ≠ 0 → Integrable fun x => g (S • x) hf : Integrable fun x => f (R • x) ⊢ Integrable f ** convert this hf (inv_ne_zero hR) ** case h.e'_5.h.h.e'_1 F : Type u_1 inst✝⁶ : NormedAddCommGroup F E : Type u_2 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℝ E inst✝³ : MeasurableSpace E inst✝² : BorelSpace E inst✝¹ : FiniteDimensional ℝ E μ : Measure E inst✝ : IsAddHaarMeasure μ f : E → F R : ℝ hR : R ≠ 0 this : ∀ {g : E → F}, Integrable g → ∀ {S : ℝ}, S ≠ 0 → Integrable fun x => g (S • x) hf : Integrable fun x => f (R • x) x✝ : E ⊢ x✝ = R • R⁻¹ • x✝ ** rw [← mul_smul, mul_inv_cancel hR, one_smul] ** Qed
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MeasureTheory.integrable_comp_mul_left_iff ** F : Type u_1 inst✝ : NormedAddCommGroup F g : ℝ → F R : ℝ hR : R ≠ 0 ⊢ (Integrable fun x => g (R * x)) ↔ Integrable g ** simpa only [smul_eq_mul] using integrable_comp_smul_iff volume g hR ** Qed
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MeasureTheory.integrable_comp_mul_right_iff ** F : Type u_1 inst✝ : NormedAddCommGroup F g : ℝ → F R : ℝ hR : R ≠ 0 ⊢ (Integrable fun x => g (x * R)) ↔ Integrable g ** simpa only [mul_comm] using integrable_comp_mul_left_iff g hR ** Qed
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Opposite.unop_injective ** α : Sort u x✝¹ x✝ : αᵒᵖ unop✝¹ unop✝ : α ⊢ { unop := unop✝¹ }.unop = { unop := unop✝ }.unop → { unop := unop✝¹ } = { unop := unop✝ } ** simp ** Qed
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MeasureTheory.limsup_measure_compl_le_of_le_liminf_measure ** Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L ⊢ limsup (fun i => ↑↑(μs i) Eᶜ) L ≤ ↑↑μ Eᶜ ** rcases L.eq_or_neBot with rfl | hne ** case inr Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L ⊢ limsup (fun i => ↑↑(μs i) Eᶜ) L ≤ ↑↑μ Eᶜ ** have meas_Ec : μ Eᶜ = 1 - μ E := by
simpa only [measure_univ] using measure_compl E_mble (measure_lt_top μ E).ne ** case inr Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L meas_Ec : ↑↑μ Eᶜ = 1 - ↑↑μ E ⊢ limsup (fun i => ↑↑(μs i) Eᶜ) L ≤ ↑↑μ Eᶜ ** have meas_i_Ec : ∀ i, μs i Eᶜ = 1 - μs i E := by
intro i
simpa only [measure_univ] using measure_compl E_mble (measure_lt_top (μs i) E).ne ** case inr Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L meas_Ec : ↑↑μ Eᶜ = 1 - ↑↑μ E meas_i_Ec : ∀ (i : ι), ↑↑(μs i) Eᶜ = 1 - ↑↑(μs i) E ⊢ limsup (fun i => ↑↑(μs i) Eᶜ) L ≤ ↑↑μ Eᶜ ** simp_rw [meas_Ec, meas_i_Ec] ** case inr Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L meas_Ec : ↑↑μ Eᶜ = 1 - ↑↑μ E meas_i_Ec : ∀ (i : ι), ↑↑(μs i) Eᶜ = 1 - ↑↑(μs i) E ⊢ limsup (fun i => 1 - ↑↑(μs i) E) L ≤ 1 - ↑↑μ E ** have obs :
(L.limsup fun i : ι => 1 - μs i E) = L.limsup ((fun x => 1 - x) ∘ fun i : ι => μs i E) := rfl ** case inr Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L meas_Ec : ↑↑μ Eᶜ = 1 - ↑↑μ E meas_i_Ec : ∀ (i : ι), ↑↑(μs i) Eᶜ = 1 - ↑↑(μs i) E obs : limsup (fun i => 1 - ↑↑(μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => ↑↑(μs i) E) L ⊢ limsup (fun i => 1 - ↑↑(μs i) E) L ≤ 1 - ↑↑μ E ** rw [obs] ** case inr Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L meas_Ec : ↑↑μ Eᶜ = 1 - ↑↑μ E meas_i_Ec : ∀ (i : ι), ↑↑(μs i) Eᶜ = 1 - ↑↑(μs i) E obs : limsup (fun i => 1 - ↑↑(μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => ↑↑(μs i) E) L ⊢ limsup ((fun x => 1 - x) ∘ fun i => ↑↑(μs i) E) L ≤ 1 - ↑↑μ E ** have := antitone_const_tsub.map_liminf_of_continuousAt (F := L)
(fun i => μs i E) (ENNReal.continuous_sub_left ENNReal.one_ne_top).continuousAt ** case inr Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L meas_Ec : ↑↑μ Eᶜ = 1 - ↑↑μ E meas_i_Ec : ∀ (i : ι), ↑↑(μs i) Eᶜ = 1 - ↑↑(μs i) E obs : limsup (fun i => 1 - ↑↑(μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => ↑↑(μs i) E) L this : 1 - liminf (fun i => ↑↑(μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => ↑↑(μs i) E) L ⊢ limsup ((fun x => 1 - x) ∘ fun i => ↑↑(μs i) E) L ≤ 1 - ↑↑μ E ** simp_rw [← this] ** case inr Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L meas_Ec : ↑↑μ Eᶜ = 1 - ↑↑μ E meas_i_Ec : ∀ (i : ι), ↑↑(μs i) Eᶜ = 1 - ↑↑(μs i) E obs : limsup (fun i => 1 - ↑↑(μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => ↑↑(μs i) E) L this : 1 - liminf (fun i => ↑↑(μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => ↑↑(μs i) E) L ⊢ 1 - liminf (fun i => ↑↑(μs i) E) L ≤ 1 - ↑↑μ E ** exact antitone_const_tsub h ** case inl Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) ⊥ ⊢ limsup (fun i => ↑↑(μs i) Eᶜ) ⊥ ≤ ↑↑μ Eᶜ ** simp only [limsup_bot, bot_le] ** Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L ⊢ ↑↑μ Eᶜ = 1 - ↑↑μ E ** simpa only [measure_univ] using measure_compl E_mble (measure_lt_top μ E).ne ** Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L meas_Ec : ↑↑μ Eᶜ = 1 - ↑↑μ E ⊢ ∀ (i : ι), ↑↑(μs i) Eᶜ = 1 - ↑↑(μs i) E ** intro i ** Ω : Type u_1 inst✝² : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω inst✝¹ : IsProbabilityMeasure μ inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i) E : Set Ω E_mble : MeasurableSet E h : ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L hne : NeBot L meas_Ec : ↑↑μ Eᶜ = 1 - ↑↑μ E i : ι ⊢ ↑↑(μs i) Eᶜ = 1 - ↑↑(μs i) E ** simpa only [measure_univ] using measure_compl E_mble (measure_lt_top (μs i) E).ne ** Qed
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MeasureTheory.tendsto_measure_of_le_liminf_measure_of_limsup_measure_le ** Ω : Type u_1 inst✝ : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω E₀ E E₁ : Set Ω E₀_subset : E₀ ⊆ E subset_E₁ : E ⊆ E₁ nulldiff : ↑↑μ (E₁ \ E₀) = 0 h_E₀ : ↑↑μ E₀ ≤ liminf (fun i => ↑↑(μs i) E₀) L h_E₁ : limsup (fun i => ↑↑(μs i) E₁) L ≤ ↑↑μ E₁ ⊢ Tendsto (fun i => ↑↑(μs i) E) L (𝓝 (↑↑μ E)) ** apply tendsto_of_le_liminf_of_limsup_le ** case hinf Ω : Type u_1 inst✝ : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω E₀ E E₁ : Set Ω E₀_subset : E₀ ⊆ E subset_E₁ : E ⊆ E₁ nulldiff : ↑↑μ (E₁ \ E₀) = 0 h_E₀ : ↑↑μ E₀ ≤ liminf (fun i => ↑↑(μs i) E₀) L h_E₁ : limsup (fun i => ↑↑(μs i) E₁) L ≤ ↑↑μ E₁ ⊢ ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L ** have E₀_ae_eq_E : E₀ =ᵐ[μ] E :=
EventuallyLE.antisymm E₀_subset.eventuallyLE
(subset_E₁.eventuallyLE.trans (ae_le_set.mpr nulldiff)) ** case hinf Ω : Type u_1 inst✝ : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω E₀ E E₁ : Set Ω E₀_subset : E₀ ⊆ E subset_E₁ : E ⊆ E₁ nulldiff : ↑↑μ (E₁ \ E₀) = 0 h_E₀ : ↑↑μ E₀ ≤ liminf (fun i => ↑↑(μs i) E₀) L h_E₁ : limsup (fun i => ↑↑(μs i) E₁) L ≤ ↑↑μ E₁ E₀_ae_eq_E : E₀ =ᵐ[μ] E ⊢ ↑↑μ E ≤ liminf (fun i => ↑↑(μs i) E) L ** calc
μ E = μ E₀ := measure_congr E₀_ae_eq_E.symm
_ ≤ L.liminf fun i => μs i E₀ := h_E₀
_ ≤ L.liminf fun i => μs i E :=
liminf_le_liminf (eventually_of_forall fun _ => measure_mono E₀_subset) ** case hsup Ω : Type u_1 inst✝ : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω E₀ E E₁ : Set Ω E₀_subset : E₀ ⊆ E subset_E₁ : E ⊆ E₁ nulldiff : ↑↑μ (E₁ \ E₀) = 0 h_E₀ : ↑↑μ E₀ ≤ liminf (fun i => ↑↑(μs i) E₀) L h_E₁ : limsup (fun i => ↑↑(μs i) E₁) L ≤ ↑↑μ E₁ ⊢ limsup (fun i => ↑↑(μs i) E) L ≤ ↑↑μ E ** have E_ae_eq_E₁ : E =ᵐ[μ] E₁ :=
EventuallyLE.antisymm subset_E₁.eventuallyLE
((ae_le_set.mpr nulldiff).trans E₀_subset.eventuallyLE) ** case hsup Ω : Type u_1 inst✝ : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω E₀ E E₁ : Set Ω E₀_subset : E₀ ⊆ E subset_E₁ : E ⊆ E₁ nulldiff : ↑↑μ (E₁ \ E₀) = 0 h_E₀ : ↑↑μ E₀ ≤ liminf (fun i => ↑↑(μs i) E₀) L h_E₁ : limsup (fun i => ↑↑(μs i) E₁) L ≤ ↑↑μ E₁ E_ae_eq_E₁ : E =ᵐ[μ] E₁ ⊢ limsup (fun i => ↑↑(μs i) E) L ≤ ↑↑μ E ** calc
(L.limsup fun i => μs i E) ≤ L.limsup fun i => μs i E₁ :=
limsup_le_limsup (eventually_of_forall fun _ => measure_mono subset_E₁)
_ ≤ μ E₁ := h_E₁
_ = μ E := measure_congr E_ae_eq_E₁.symm ** case h Ω : Type u_1 inst✝ : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω E₀ E E₁ : Set Ω E₀_subset : E₀ ⊆ E subset_E₁ : E ⊆ E₁ nulldiff : ↑↑μ (E₁ \ E₀) = 0 h_E₀ : ↑↑μ E₀ ≤ liminf (fun i => ↑↑(μs i) E₀) L h_E₁ : limsup (fun i => ↑↑(μs i) E₁) L ≤ ↑↑μ E₁ ⊢ autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) L fun i => ↑↑(μs i) E) _auto✝ ** infer_param ** case h' Ω : Type u_1 inst✝ : MeasurableSpace Ω ι : Type u_2 L : Filter ι μ : Measure Ω μs : ι → Measure Ω E₀ E E₁ : Set Ω E₀_subset : E₀ ⊆ E subset_E₁ : E ⊆ E₁ nulldiff : ↑↑μ (E₁ \ E₀) = 0 h_E₀ : ↑↑μ E₀ ≤ liminf (fun i => ↑↑(μs i) E₀) L h_E₁ : limsup (fun i => ↑↑(μs i) E₁) L ≤ ↑↑μ E₁ ⊢ autoParam (IsBoundedUnder (fun x x_1 => x ≥ x_1) L fun i => ↑↑(μs i) E) _auto✝ ** infer_param ** Qed
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MeasureTheory.measure_of_cont_bdd_of_tendsto_filter_indicator ** Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω ι : Type u_2 L : Filter ι inst✝³ : IsCountablyGenerated L inst✝² : TopologicalSpace Ω inst✝¹ : OpensMeasurableSpace Ω μ : Measure Ω inst✝ : IsFiniteMeasure μ c : ℝ≥0 E : Set Ω E_mble : MeasurableSet E fs : ι → Ω →ᵇ ℝ≥0 fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (indicator E (fun x => 1) ω)) ⊢ Tendsto (fun n => ∫⁻ (ω : Ω), ↑(↑(fs n) ω) ∂μ) L (𝓝 (↑↑μ E)) ** convert FiniteMeasure.tendsto_lintegral_nn_filter_of_le_const μ fs_bdd fs_lim ** case h.e'_5.h.e'_3 Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω ι : Type u_2 L : Filter ι inst✝³ : IsCountablyGenerated L inst✝² : TopologicalSpace Ω inst✝¹ : OpensMeasurableSpace Ω μ : Measure Ω inst✝ : IsFiniteMeasure μ c : ℝ≥0 E : Set Ω E_mble : MeasurableSet E fs : ι → Ω →ᵇ ℝ≥0 fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (indicator E (fun x => 1) ω)) ⊢ ↑↑μ E = ∫⁻ (ω : Ω), ↑(indicator E (fun x => 1) ω) ∂μ ** have aux : ∀ ω, indicator E (fun _ => (1 : ℝ≥0∞)) ω = ↑(indicator E (fun _ => (1 : ℝ≥0)) ω) :=
fun ω => by simp only [ENNReal.coe_indicator, ENNReal.coe_one] ** case h.e'_5.h.e'_3 Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω ι : Type u_2 L : Filter ι inst✝³ : IsCountablyGenerated L inst✝² : TopologicalSpace Ω inst✝¹ : OpensMeasurableSpace Ω μ : Measure Ω inst✝ : IsFiniteMeasure μ c : ℝ≥0 E : Set Ω E_mble : MeasurableSet E fs : ι → Ω →ᵇ ℝ≥0 fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (indicator E (fun x => 1) ω)) aux : ∀ (ω : Ω), indicator E (fun x => 1) ω = ↑(indicator E (fun x => 1) ω) ⊢ ↑↑μ E = ∫⁻ (ω : Ω), ↑(indicator E (fun x => 1) ω) ∂μ ** simp_rw [← aux, lintegral_indicator _ E_mble] ** case h.e'_5.h.e'_3 Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω ι : Type u_2 L : Filter ι inst✝³ : IsCountablyGenerated L inst✝² : TopologicalSpace Ω inst✝¹ : OpensMeasurableSpace Ω μ : Measure Ω inst✝ : IsFiniteMeasure μ c : ℝ≥0 E : Set Ω E_mble : MeasurableSet E fs : ι → Ω →ᵇ ℝ≥0 fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (indicator E (fun x => 1) ω)) aux : ∀ (ω : Ω), indicator E (fun x => 1) ω = ↑(indicator E (fun x => 1) ω) ⊢ ↑↑μ E = ∫⁻ (x : Ω) in E, 1 ∂μ ** simp only [lintegral_one, Measure.restrict_apply, MeasurableSet.univ, univ_inter] ** Ω : Type u_1 inst✝⁴ : MeasurableSpace Ω ι : Type u_2 L : Filter ι inst✝³ : IsCountablyGenerated L inst✝² : TopologicalSpace Ω inst✝¹ : OpensMeasurableSpace Ω μ : Measure Ω inst✝ : IsFiniteMeasure μ c : ℝ≥0 E : Set Ω E_mble : MeasurableSet E fs : ι → Ω →ᵇ ℝ≥0 fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (indicator E (fun x => 1) ω)) ω : Ω ⊢ indicator E (fun x => 1) ω = ↑(indicator E (fun x => 1) ω) ** simp only [ENNReal.coe_indicator, ENNReal.coe_one] ** Qed
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OrthonormalBasis.volume_parallelepiped ** ι : Type u_1 F : Type u_2 inst✝⁵ : Fintype ι inst✝⁴ : NormedAddCommGroup F inst✝³ : InnerProductSpace ℝ F inst✝² : FiniteDimensional ℝ F inst✝¹ : MeasurableSpace F inst✝ : BorelSpace F b : OrthonormalBasis ι ℝ F ⊢ ↑↑volume (parallelepiped ↑b) = 1 ** haveI : Fact (finrank ℝ F = finrank ℝ F) := ⟨rfl⟩ ** ι : Type u_1 F : Type u_2 inst✝⁵ : Fintype ι inst✝⁴ : NormedAddCommGroup F inst✝³ : InnerProductSpace ℝ F inst✝² : FiniteDimensional ℝ F inst✝¹ : MeasurableSpace F inst✝ : BorelSpace F b : OrthonormalBasis ι ℝ F this : Fact (finrank ℝ F = finrank ℝ F) ⊢ ↑↑volume (parallelepiped ↑b) = 1 ** let o := (stdOrthonormalBasis ℝ F).toBasis.orientation ** ι : Type u_1 F : Type u_2 inst✝⁵ : Fintype ι inst✝⁴ : NormedAddCommGroup F inst✝³ : InnerProductSpace ℝ F inst✝² : FiniteDimensional ℝ F inst✝¹ : MeasurableSpace F inst✝ : BorelSpace F b : OrthonormalBasis ι ℝ F this : Fact (finrank ℝ F = finrank ℝ F) o : Orientation ℝ F (Fin (finrank ℝ F)) := Basis.orientation (OrthonormalBasis.toBasis (stdOrthonormalBasis ℝ F)) ⊢ ↑↑volume (parallelepiped ↑b) = 1 ** rw [← o.measure_eq_volume] ** ι : Type u_1 F : Type u_2 inst✝⁵ : Fintype ι inst✝⁴ : NormedAddCommGroup F inst✝³ : InnerProductSpace ℝ F inst✝² : FiniteDimensional ℝ F inst✝¹ : MeasurableSpace F inst✝ : BorelSpace F b : OrthonormalBasis ι ℝ F this : Fact (finrank ℝ F = finrank ℝ F) o : Orientation ℝ F (Fin (finrank ℝ F)) := Basis.orientation (OrthonormalBasis.toBasis (stdOrthonormalBasis ℝ F)) ⊢ ↑↑(AlternatingMap.measure (Orientation.volumeForm o)) (parallelepiped ↑b) = 1 ** exact o.measure_orthonormalBasis b ** Qed
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OrthonormalBasis.addHaar_eq_volume ** ι✝ : Type u_1 F✝ : Type u_2 inst✝¹¹ : Fintype ι✝ inst✝¹⁰ : NormedAddCommGroup F✝ inst✝⁹ : InnerProductSpace ℝ F✝ inst✝⁸ : FiniteDimensional ℝ F✝ inst✝⁷ : MeasurableSpace F✝ inst✝⁶ : BorelSpace F✝ ι : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : NormedAddCommGroup F inst✝³ : InnerProductSpace ℝ F inst✝² : FiniteDimensional ℝ F inst✝¹ : MeasurableSpace F inst✝ : BorelSpace F b : OrthonormalBasis ι ℝ F ⊢ Basis.addHaar (OrthonormalBasis.toBasis b) = volume ** rw [Basis.addHaar_eq_iff] ** ι✝ : Type u_1 F✝ : Type u_2 inst✝¹¹ : Fintype ι✝ inst✝¹⁰ : NormedAddCommGroup F✝ inst✝⁹ : InnerProductSpace ℝ F✝ inst✝⁸ : FiniteDimensional ℝ F✝ inst✝⁷ : MeasurableSpace F✝ inst✝⁶ : BorelSpace F✝ ι : Type u_3 F : Type u_4 inst✝⁵ : Fintype ι inst✝⁴ : NormedAddCommGroup F inst✝³ : InnerProductSpace ℝ F inst✝² : FiniteDimensional ℝ F inst✝¹ : MeasurableSpace F inst✝ : BorelSpace F b : OrthonormalBasis ι ℝ F ⊢ ↑↑volume ↑(Basis.parallelepiped (OrthonormalBasis.toBasis b)) = 1 ** exact b.volume_parallelepiped ** Qed
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Tree.numLeaves_eq_numNodes_succ ** α : Type u x : Tree α ⊢ numLeaves x = numNodes x + 1 ** induction x <;> simp [*, Nat.add_comm, Nat.add_assoc, Nat.add_left_comm] ** Qed
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Tree.left_node_right_eq_self ** α : Type u h : nil ≠ nil ⊢ node () (left nil) (right nil) = nil ** trivial ** Qed
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rieszContentAux_image_nonempty ** X : Type u_1 inst✝ : TopologicalSpace X Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 K : Compacts X ⊢ Set.Nonempty (↑Λ '' {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x}) ** rw [nonempty_image_iff] ** X : Type u_1 inst✝ : TopologicalSpace X Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 K : Compacts X ⊢ Set.Nonempty {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x} ** use (1 : X →ᵇ ℝ≥0) ** case h X : Type u_1 inst✝ : TopologicalSpace X Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 K : Compacts X ⊢ 1 ∈ {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x} ** intro x _ ** case h X : Type u_1 inst✝ : TopologicalSpace X Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 K : Compacts X x : X a✝ : x ∈ K ⊢ 1 ≤ ↑1 x ** simp only [BoundedContinuousFunction.coe_one, Pi.one_apply] ** case h X : Type u_1 inst✝ : TopologicalSpace X Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 K : Compacts X x : X a✝ : x ∈ K ⊢ 1 ≤ 1 ** rfl ** Qed
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exists_lt_rieszContentAux_add_pos ** X : Type u_1 inst✝ : TopologicalSpace X Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 K : Compacts X ε : ℝ≥0 εpos : 0 < ε ⊢ ∃ f, (∀ (x : X), x ∈ K → 1 ≤ ↑f x) ∧ ↑Λ f < rieszContentAux Λ K + ε ** obtain ⟨α, ⟨⟨f, f_hyp⟩, α_hyp⟩⟩ :=
exists_lt_of_csInf_lt (rieszContentAux_image_nonempty Λ K)
(lt_add_of_pos_right (rieszContentAux Λ K) εpos) ** case intro.intro.intro X : Type u_1 inst✝ : TopologicalSpace X Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 K : Compacts X ε : ℝ≥0 εpos : 0 < ε α : ℝ≥0 α_hyp : α < rieszContentAux Λ K + ε f : X →ᵇ ℝ≥0 f_hyp : f ∈ {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x} ∧ ↑Λ f = α ⊢ ∃ f, (∀ (x : X), x ∈ K → 1 ≤ ↑f x) ∧ ↑Λ f < rieszContentAux Λ K + ε ** refine' ⟨f, f_hyp.left, _⟩ ** case intro.intro.intro X : Type u_1 inst✝ : TopologicalSpace X Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 K : Compacts X ε : ℝ≥0 εpos : 0 < ε α : ℝ≥0 α_hyp : α < rieszContentAux Λ K + ε f : X →ᵇ ℝ≥0 f_hyp : f ∈ {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x} ∧ ↑Λ f = α ⊢ ↑Λ f < rieszContentAux Λ K + ε ** rw [f_hyp.right] ** case intro.intro.intro X : Type u_1 inst✝ : TopologicalSpace X Λ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0 K : Compacts X ε : ℝ≥0 εpos : 0 < ε α : ℝ≥0 α_hyp : α < rieszContentAux Λ K + ε f : X →ᵇ ℝ≥0 f_hyp : f ∈ {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x} ∧ ↑Λ f = α ⊢ α < rieszContentAux Λ K + ε ** exact α_hyp ** Qed
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Real.volume_Ico ** ι : Type u_1 inst✝ : Fintype ι a b : ℝ ⊢ ↑↑volume (Ico a b) = ofReal (b - a) ** simp [volume_val] ** Qed
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Real.volume_Icc ** ι : Type u_1 inst✝ : Fintype ι a b : ℝ ⊢ ↑↑volume (Icc a b) = ofReal (b - a) ** simp [volume_val] ** Qed
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Real.volume_Ioo ** ι : Type u_1 inst✝ : Fintype ι a b : ℝ ⊢ ↑↑volume (Ioo a b) = ofReal (b - a) ** simp [volume_val] ** Qed
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Real.volume_Ioc ** ι : Type u_1 inst✝ : Fintype ι a b : ℝ ⊢ ↑↑volume (Ioc a b) = ofReal (b - a) ** simp [volume_val] ** Qed
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Real.volume_singleton ** ι : Type u_1 inst✝ : Fintype ι a : ℝ ⊢ ↑↑volume {a} = 0 ** simp [volume_val] ** Qed
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Real.volume_univ ** ι : Type u_1 inst✝ : Fintype ι r : ℝ≥0 ⊢ ↑r = ↑↑volume (Icc 0 ↑r) ** simp ** Qed
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Real.volume_ball ** ι : Type u_1 inst✝ : Fintype ι a r : ℝ ⊢ ↑↑volume (Metric.ball a r) = ofReal (2 * r) ** rw [ball_eq_Ioo, volume_Ioo, ← sub_add, add_sub_cancel', two_mul] ** Qed
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Real.volume_closedBall ** ι : Type u_1 inst✝ : Fintype ι a r : ℝ ⊢ ↑↑volume (Metric.closedBall a r) = ofReal (2 * r) ** rw [closedBall_eq_Icc, volume_Icc, ← sub_add, add_sub_cancel', two_mul] ** Qed
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Real.volume_interval ** ι : Type u_1 inst✝ : Fintype ι a b : ℝ ⊢ ↑↑volume (uIcc a b) = ofReal |b - a| ** rw [← Icc_min_max, volume_Icc, max_sub_min_eq_abs] ** Qed
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Real.volume_Ioi ** ι : Type u_1 inst✝ : Fintype ι a : ℝ n : ℕ ⊢ ↑n = ↑↑volume (Ioo a (a + ↑n)) ** simp ** Qed
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Real.volume_Ici ** ι : Type u_1 inst✝ : Fintype ι a : ℝ ⊢ ↑↑volume (Ici a) = ⊤ ** rw [← measure_congr Ioi_ae_eq_Ici] ** ι : Type u_1 inst✝ : Fintype ι a : ℝ ⊢ ↑↑volume (Ioi a) = ⊤ ** simp ** Qed
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Real.volume_Iio ** ι : Type u_1 inst✝ : Fintype ι a : ℝ n : ℕ ⊢ ↑n = ↑↑volume (Ioo (a - ↑n) a) ** simp ** Qed
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Real.volume_Iic ** ι : Type u_1 inst✝ : Fintype ι a : ℝ ⊢ ↑↑volume (Iic a) = ⊤ ** rw [← measure_congr Iio_ae_eq_Iic] ** ι : Type u_1 inst✝ : Fintype ι a : ℝ ⊢ ↑↑volume (Iio a) = ⊤ ** simp ** Qed
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Real.volume_le_diam ** ι : Type u_1 inst✝ : Fintype ι s : Set ℝ ⊢ ↑↑volume s ≤ EMetric.diam s ** by_cases hs : Bornology.IsBounded s ** case pos ι : Type u_1 inst✝ : Fintype ι s : Set ℝ hs : Bornology.IsBounded s ⊢ ↑↑volume s ≤ EMetric.diam s ** rw [Real.ediam_eq hs, ← volume_Icc] ** case pos ι : Type u_1 inst✝ : Fintype ι s : Set ℝ hs : Bornology.IsBounded s ⊢ ↑↑volume s ≤ ↑↑volume (Icc (sInf s) (sSup s)) ** exact volume.mono (Real.subset_Icc_sInf_sSup_of_isBounded hs) ** case neg ι : Type u_1 inst✝ : Fintype ι s : Set ℝ hs : ¬Bornology.IsBounded s ⊢ ↑↑volume s ≤ EMetric.diam s ** rw [Metric.ediam_of_unbounded hs] ** case neg ι : Type u_1 inst✝ : Fintype ι s : Set ℝ hs : ¬Bornology.IsBounded s ⊢ ↑↑volume s ≤ ⊤ ** exact le_top ** Qed
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Filter.Eventually.volume_pos_of_nhds_real ** ι : Type u_1 inst✝ : Fintype ι p : ℝ → Prop a : ℝ h : ∀ᶠ (x : ℝ) in 𝓝 a, p x ⊢ 0 < ↑↑volume {x | p x} ** rcases h.exists_Ioo_subset with ⟨l, u, hx, hs⟩ ** case intro.intro.intro ι : Type u_1 inst✝ : Fintype ι p : ℝ → Prop a : ℝ h : ∀ᶠ (x : ℝ) in 𝓝 a, p x l u : ℝ hx : a ∈ Ioo l u hs : Ioo l u ⊆ {x | p x} ⊢ 0 < ↑↑volume {x | p x} ** refine' lt_of_lt_of_le _ (measure_mono hs) ** case intro.intro.intro ι : Type u_1 inst✝ : Fintype ι p : ℝ → Prop a : ℝ h : ∀ᶠ (x : ℝ) in 𝓝 a, p x l u : ℝ hx : a ∈ Ioo l u hs : Ioo l u ⊆ {x | p x} ⊢ 0 < ↑↑volume (Ioo l u) ** simpa [-mem_Ioo] using hx.1.trans hx.2 ** Qed
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Real.volume_Icc_pi ** ι : Type u_1 inst✝ : Fintype ι a b : ι → ℝ ⊢ ↑↑volume (Icc a b) = ∏ i : ι, ofReal (b i - a i) ** rw [← pi_univ_Icc, volume_pi_pi] ** ι : Type u_1 inst✝ : Fintype ι a b : ι → ℝ ⊢ ∏ i : ι, ↑↑volume (Icc (a i) (b i)) = ∏ i : ι, ofReal (b i - a i) ** simp only [Real.volume_Icc] ** Qed
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Real.volume_Icc_pi_toReal ** ι : Type u_1 inst✝ : Fintype ι a b : ι → ℝ h : a ≤ b ⊢ ENNReal.toReal (↑↑volume (Icc a b)) = ∏ i : ι, (b i - a i) ** simp only [volume_Icc_pi, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))] ** Qed
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Real.map_volume_mul_left ** ι : Type u_1 inst✝ : Fintype ι a : ℝ h : a ≠ 0 ⊢ Measure.map (fun x => a * x) volume = ofReal |a⁻¹| • volume ** conv_rhs =>
rw [← Real.smul_map_volume_mul_left h, smul_smul, ← ENNReal.ofReal_mul (abs_nonneg _), ←
abs_mul, inv_mul_cancel h, abs_one, ENNReal.ofReal_one, one_smul] ** Qed
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Real.volume_preimage_mul_left ** ι : Type u_1 inst✝ : Fintype ι a : ℝ h : a ≠ 0 s : Set ℝ ⊢ ↑↑(Measure.map (fun x => a * x) volume) s = ofReal |a⁻¹| * ↑↑volume s ** rw [map_volume_mul_left h] ** ι : Type u_1 inst✝ : Fintype ι a : ℝ h : a ≠ 0 s : Set ℝ ⊢ ↑↑(ofReal |a⁻¹| • volume) s = ofReal |a⁻¹| * ↑↑volume s ** rfl ** Qed
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Real.smul_map_volume_mul_right ** ι : Type u_1 inst✝ : Fintype ι a : ℝ h : a ≠ 0 ⊢ ofReal |a| • Measure.map (fun x => x * a) volume = volume ** simpa only [mul_comm] using Real.smul_map_volume_mul_left h ** Qed
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Real.map_volume_mul_right ** ι : Type u_1 inst✝ : Fintype ι a : ℝ h : a ≠ 0 ⊢ Measure.map (fun x => x * a) volume = ofReal |a⁻¹| • volume ** simpa only [mul_comm] using Real.map_volume_mul_left h ** Qed
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regionBetween_subset ** α : Type u_1 f g : α → ℝ s : Set α ⊢ regionBetween f g s ⊆ s ×ˢ univ ** simpa only [prod_univ, regionBetween, Set.preimage, setOf_subset_setOf] using fun a => And.left ** Qed
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measurableSet_regionBetween ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet (regionBetween f g s) ** dsimp only [regionBetween, Ioo, mem_setOf_eq, setOf_and] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet ({a | a.1 ∈ s} ∩ {a | a.2 ∈ {a_1 | f a.1 < a_1} ∩ {a_1 | a_1 < g a.1}}) ** refine'
MeasurableSet.inter _
((measurableSet_lt (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_lt measurable_snd (hg.comp measurable_fst))) ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet {a | a.1 ∈ s} ** exact measurable_fst hs ** Qed
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measurableSet_region_between_oc ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Ioc (f p.1) (g p.1)} ** dsimp only [regionBetween, Ioc, mem_setOf_eq, setOf_and] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet ({a | a.1 ∈ s} ∩ {a | a.2 ∈ {a_1 | f a.1 < a_1} ∩ {a_1 | a_1 ≤ g a.1}}) ** refine'
MeasurableSet.inter _
((measurableSet_lt (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_le measurable_snd (hg.comp measurable_fst))) ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet {a | a.1 ∈ s} ** exact measurable_fst hs ** Qed
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measurableSet_region_between_co ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Ico (f p.1) (g p.1)} ** dsimp only [regionBetween, Ico, mem_setOf_eq, setOf_and] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet ({a | a.1 ∈ s} ∩ {a | a.2 ∈ {a_1 | f a.1 ≤ a_1} ∩ {a_1 | a_1 < g a.1}}) ** refine'
MeasurableSet.inter _
((measurableSet_le (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_lt measurable_snd (hg.comp measurable_fst))) ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet {a | a.1 ∈ s} ** exact measurable_fst hs ** Qed
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measurableSet_region_between_cc ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Icc (f p.1) (g p.1)} ** dsimp only [regionBetween, Icc, mem_setOf_eq, setOf_and] ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet ({a | a.1 ∈ s} ∩ {a | a.2 ∈ {a_1 | f a.1 ≤ a_1} ∩ {a_1 | a_1 ≤ g a.1}}) ** refine'
MeasurableSet.inter _
((measurableSet_le (hf.comp measurable_fst) measurable_snd).inter
(measurableSet_le measurable_snd (hg.comp measurable_fst))) ** α : Type u_1 inst✝ : MeasurableSpace α μ : Measure α f g : α → ℝ s : Set α hf : Measurable f hg : Measurable g hs : MeasurableSet s ⊢ MeasurableSet {a | a.1 ∈ s} ** exact measurable_fst hs ** Qed
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ae_of_mem_of_ae_of_mem_inter_Ioo ** μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s → p x ** let T : s × s → Set ℝ := fun p => Ioo p.1 p.2 ** μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s → p x ** let u := ⋃ i : ↥s × ↥s, T i ** μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s → p x ** have hfinite : (s \ u).Finite := s.finite_diff_iUnion_Ioo' ** μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s → p x ** obtain ⟨A, A_count, hA⟩ :
∃ A : Set (↥s × ↥s), A.Countable ∧ ⋃ i ∈ A, T i = ⋃ i : ↥s × ↥s, T i :=
isOpen_iUnion_countable _ fun p => isOpen_Ioo ** case intro.intro μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s → p x ** have M : ∀ᵐ x ∂μ, x ∉ s \ u := hfinite.countable.ae_not_mem _ ** case intro.intro μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u M' : ∀ᵐ (x : ℝ) ∂μ, ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s → p x ** filter_upwards [M, M'] with x hx h'x ** case h μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u M' : ∀ᵐ (x : ℝ) ∂μ, ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x x : ℝ hx : ¬x ∈ s \ u h'x : ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x ⊢ x ∈ s → p x ** intro xs ** case h μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u M' : ∀ᵐ (x : ℝ) ∂μ, ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x x : ℝ hx : ¬x ∈ s \ u h'x : ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x xs : x ∈ s ⊢ p x ** by_cases Hx : x ∈ ⋃ i : ↥s × ↥s, T i ** μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u ⊢ ∀ᵐ (x : ℝ) ∂μ, ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x ** rw [ae_ball_iff A_count] ** μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u ⊢ ∀ (i : ↑s × ↑s), i ∈ A → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ T i → p x ** rintro ⟨⟨a, as⟩, ⟨b, bs⟩⟩ - ** case mk.mk.mk μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u a : ℝ as : a ∈ s b : ℝ bs : b ∈ s ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ T ({ val := a, property := as }, { val := b, property := bs }) → p x ** change ∀ᵐ x : ℝ ∂μ, x ∈ s ∩ Ioo a b → p x ** case mk.mk.mk μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u a : ℝ as : a ∈ s b : ℝ bs : b ∈ s ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x ** rcases le_or_lt b a with (hba | hab) ** case mk.mk.mk.inl μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u a : ℝ as : a ∈ s b : ℝ bs : b ∈ s hba : b ≤ a ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x ** simp only [Ioo_eq_empty_of_le hba, inter_empty, IsEmpty.forall_iff, eventually_true,
mem_empty_iff_false] ** case mk.mk.mk.inr μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u a : ℝ as : a ∈ s b : ℝ bs : b ∈ s hab : a < b ⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x ** exact h a b as bs hab ** case pos μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u M' : ∀ᵐ (x : ℝ) ∂μ, ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x x : ℝ hx : ¬x ∈ s \ u h'x : ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x xs : x ∈ s Hx : x ∈ ⋃ i, T i ⊢ p x ** rw [← hA] at Hx ** case pos μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u M' : ∀ᵐ (x : ℝ) ∂μ, ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x x : ℝ hx : ¬x ∈ s \ u h'x : ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x xs : x ∈ s Hx : x ∈ ⋃ i ∈ A, T i ⊢ p x ** obtain ⟨p, pA, xp⟩ : ∃ p : ↥s × ↥s, p ∈ A ∧ x ∈ T p := by
simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx ** case pos.intro.intro μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p✝ : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p✝ x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u M' : ∀ᵐ (x : ℝ) ∂μ, ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p✝ x x : ℝ hx : ¬x ∈ s \ u h'x : ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p✝ x xs : x ∈ s Hx : x ∈ ⋃ i ∈ A, T i p : ↑s × ↑s pA : p ∈ A xp : x ∈ T p ⊢ p✝ x ** apply h'x p pA ⟨xs, xp⟩ ** μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u M' : ∀ᵐ (x : ℝ) ∂μ, ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x x : ℝ hx : ¬x ∈ s \ u h'x : ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x xs : x ∈ s Hx : x ∈ ⋃ i ∈ A, T i ⊢ ∃ p, p ∈ A ∧ x ∈ T p ** simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx ** case neg μ : Measure ℝ inst✝ : NoAtoms μ s : Set ℝ p : ℝ → Prop h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2 u : Set ℝ := ⋃ i, T i hfinite : Set.Finite (s \ u) A : Set (↑s × ↑s) A_count : Set.Countable A hA : ⋃ i ∈ A, T i = ⋃ i, T i M : ∀ᵐ (x : ℝ) ∂μ, ¬x ∈ s \ u M' : ∀ᵐ (x : ℝ) ∂μ, ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x x : ℝ hx : ¬x ∈ s \ u h'x : ∀ (i : ↑s × ↑s), i ∈ A → x ∈ s ∩ T i → p x xs : x ∈ s Hx : ¬x ∈ ⋃ i, T i ⊢ p x ** exact False.elim (hx ⟨xs, Hx⟩) ** Qed
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MeasureTheory.Memℒp.snorm_mk_lt_top ** α✝ : Type u_1 E✝ : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α✝ p✝ : ℝ≥0∞ q : ℝ μ✝ ν : Measure α✝ inst✝⁴ : NormedAddCommGroup E✝ inst✝³ : NormedAddCommGroup F inst✝² : NormedAddCommGroup G α : Type u_5 E : Type u_6 inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : NormedAddCommGroup E p : ℝ≥0∞ f : α → E hfp : Memℒp f p ⊢ snorm (↑(AEEqFun.mk f (_ : AEStronglyMeasurable f μ))) p μ < ⊤ ** simp [hfp.2] ** Qed
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MeasureTheory.Memℒp.toLp_congr ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f g : α → E hf : Memℒp f p hg : Memℒp g p hfg : f =ᵐ[μ] g ⊢ toLp f hf = toLp g hg ** simp [toLp, hfg] ** Qed
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MeasureTheory.Memℒp.toLp_eq_toLp_iff ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f g : α → E hf : Memℒp f p hg : Memℒp g p ⊢ toLp f hf = toLp g hg ↔ f =ᵐ[μ] g ** simp [toLp] ** Qed
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MeasureTheory.Lp.ext ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f g : { x // x ∈ Lp E p } h : ↑↑f =ᵐ[μ] ↑↑g ⊢ f = g ** cases f ** case mk α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G g : { x // x ∈ Lp E p } val✝ : α →ₘ[μ] E property✝ : val✝ ∈ Lp E p h : ↑↑{ val := val✝, property := property✝ } =ᵐ[μ] ↑↑g ⊢ { val := val✝, property := property✝ } = g ** cases g ** case mk.mk α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G val✝¹ : α →ₘ[μ] E property✝¹ : val✝¹ ∈ Lp E p val✝ : α →ₘ[μ] E property✝ : val✝ ∈ Lp E p h : ↑↑{ val := val✝¹, property := property✝¹ } =ᵐ[μ] ↑↑{ val := val✝, property := property✝ } ⊢ { val := val✝¹, property := property✝¹ } = { val := val✝, property := property✝ } ** simp only [Subtype.mk_eq_mk] ** case mk.mk α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G val✝¹ : α →ₘ[μ] E property✝¹ : val✝¹ ∈ Lp E p val✝ : α →ₘ[μ] E property✝ : val✝ ∈ Lp E p h : ↑↑{ val := val✝¹, property := property✝¹ } =ᵐ[μ] ↑↑{ val := val✝, property := property✝ } ⊢ val✝¹ = val✝ ** exact AEEqFun.ext h ** Qed
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MeasureTheory.Lp.mem_Lp_iff_memℒp ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : α →ₘ[μ] E ⊢ f ∈ Lp E p ↔ Memℒp (↑f) p ** simp [mem_Lp_iff_snorm_lt_top, Memℒp, f.stronglyMeasurable.aestronglyMeasurable] ** Qed
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MeasureTheory.Lp.toLp_coeFn ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } hf : Memℒp (↑↑f) p ⊢ Memℒp.toLp (↑↑f) hf = f ** cases f ** case mk α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G val✝ : α →ₘ[μ] E property✝ : val✝ ∈ Lp E p hf : Memℒp (↑↑{ val := val✝, property := property✝ }) p ⊢ Memℒp.toLp (↑↑{ val := val✝, property := property✝ }) hf = { val := val✝, property := property✝ } ** simp [Memℒp.toLp] ** Qed
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MeasureTheory.Lp.norm_toLp ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : α → E hf : Memℒp f p ⊢ ‖Memℒp.toLp f hf‖ = ENNReal.toReal (snorm f p μ) ** erw [norm_def, snorm_congr_ae (Memℒp.coeFn_toLp hf)] ** Qed
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MeasureTheory.Lp.dist_def ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f g : { x // x ∈ Lp E p } ⊢ dist f g = ENNReal.toReal (snorm (↑↑f - ↑↑g) p μ) ** simp_rw [dist, norm_def] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f g : { x // x ∈ Lp E p } ⊢ ENNReal.toReal (snorm (↑↑(f - g)) p μ) = ENNReal.toReal (snorm (↑↑f - ↑↑g) p μ) ** refine congr_arg _ ?_ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f g : { x // x ∈ Lp E p } ⊢ snorm (↑↑(f - g)) p μ = snorm (↑↑f - ↑↑g) p μ ** apply snorm_congr_ae (coeFn_sub _ _) ** Qed
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MeasureTheory.Lp.edist_dist ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f g : { x // x ∈ Lp E p } ⊢ edist f g = ENNReal.ofReal (dist f g) ** rw [edist_def, dist_def, ← snorm_congr_ae (coeFn_sub _ _),
ENNReal.ofReal_toReal (snorm_ne_top (f - g))] ** Qed
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MeasureTheory.Lp.edist_toLp_toLp ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f g : α → E hf : Memℒp f p hg : Memℒp g p ⊢ edist (Memℒp.toLp f hf) (Memℒp.toLp g hg) = snorm (f - g) p μ ** rw [edist_def] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f g : α → E hf : Memℒp f p hg : Memℒp g p ⊢ snorm (↑↑(Memℒp.toLp f hf) - ↑↑(Memℒp.toLp g hg)) p μ = snorm (f - g) p μ ** exact snorm_congr_ae (hf.coeFn_toLp.sub hg.coeFn_toLp) ** Qed
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MeasureTheory.Lp.edist_toLp_zero ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : α → E hf : Memℒp f p ⊢ edist (Memℒp.toLp f hf) 0 = snorm f p μ ** convert edist_toLp_toLp f 0 hf zero_memℒp ** case h.e'_3.h.e'_5 α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : α → E hf : Memℒp f p ⊢ f = f - 0 ** simp ** Qed
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MeasureTheory.Lp.nnnorm_zero ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G ⊢ ‖0‖₊ = 0 ** rw [nnnorm_def] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G ⊢ ENNReal.toNNReal (snorm (↑↑0) p μ) = 0 ** change (snorm (⇑(0 : α →ₘ[μ] E)) p μ).toNNReal = 0 ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G ⊢ ENNReal.toNNReal (snorm (↑0) p μ) = 0 ** simp [snorm_congr_ae AEEqFun.coeFn_zero, snorm_zero] ** Qed
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MeasureTheory.Lp.norm_measure_zero ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } ⊢ ‖f‖ = 0 ** simp [norm_def] ** Qed
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MeasureTheory.Lp.norm_exponent_zero ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E 0 } ⊢ ‖f‖ = 0 ** simp [norm_def] ** Qed
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MeasureTheory.Lp.nnnorm_eq_zero_iff ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } hp : 0 < p ⊢ ‖f‖₊ = 0 ↔ f = 0 ** refine' ⟨fun hf => _, fun hf => by simp [hf]⟩ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } hp : 0 < p hf : ‖f‖₊ = 0 ⊢ f = 0 ** rw [nnnorm_def, ENNReal.toNNReal_eq_zero_iff] at hf ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } hp : 0 < p hf : snorm (↑↑f) p μ = 0 ∨ snorm (↑↑f) p μ = ⊤ ⊢ f = 0 ** cases hf with
| inl hf =>
rw [snorm_eq_zero_iff (Lp.aestronglyMeasurable f) hp.ne.symm] at hf
exact Subtype.eq (AEEqFun.ext (hf.trans AEEqFun.coeFn_zero.symm))
| inr hf =>
exact absurd hf (snorm_ne_top f) ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } hp : 0 < p hf : f = 0 ⊢ ‖f‖₊ = 0 ** simp [hf] ** case inl α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } hp : 0 < p hf : snorm (↑↑f) p μ = 0 ⊢ f = 0 ** rw [snorm_eq_zero_iff (Lp.aestronglyMeasurable f) hp.ne.symm] at hf ** case inl α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } hp : 0 < p hf : ↑↑f =ᵐ[μ] 0 ⊢ f = 0 ** exact Subtype.eq (AEEqFun.ext (hf.trans AEEqFun.coeFn_zero.symm)) ** case inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } hp : 0 < p hf : snorm (↑↑f) p μ = ⊤ ⊢ f = 0 ** exact absurd hf (snorm_ne_top f) ** Qed
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MeasureTheory.Lp.nnnorm_neg ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } ⊢ ‖-f‖₊ = ‖f‖₊ ** rw [nnnorm_def, nnnorm_def, snorm_congr_ae (coeFn_neg _), snorm_neg] ** Qed
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MeasureTheory.Lp.nnnorm_le_mul_nnnorm_of_ae_le_mul ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : ℝ≥0 f : { x // x ∈ Lp E p } g : { x // x ∈ Lp F p } h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊ ⊢ ‖f‖₊ ≤ c * ‖g‖₊ ** simp only [nnnorm_def] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : ℝ≥0 f : { x // x ∈ Lp E p } g : { x // x ∈ Lp F p } h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊ ⊢ ENNReal.toNNReal (snorm (↑↑f) p μ) ≤ c * ENNReal.toNNReal (snorm (↑↑g) p μ) ** have := snorm_le_nnreal_smul_snorm_of_ae_le_mul h p ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : ℝ≥0 f : { x // x ∈ Lp E p } g : { x // x ∈ Lp F p } h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊ this : snorm (fun x => ↑↑f x) p μ ≤ c • snorm (fun x => ↑↑g x) p μ ⊢ ENNReal.toNNReal (snorm (↑↑f) p μ) ≤ c * ENNReal.toNNReal (snorm (↑↑g) p μ) ** rwa [← ENNReal.toNNReal_le_toNNReal, ENNReal.smul_def, smul_eq_mul, ENNReal.toNNReal_mul,
ENNReal.toNNReal_coe] at this ** case ha α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : ℝ≥0 f : { x // x ∈ Lp E p } g : { x // x ∈ Lp F p } h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊ this : snorm (fun x => ↑↑f x) p μ ≤ c • snorm (fun x => ↑↑g x) p μ ⊢ snorm (fun x => ↑↑f x) p μ ≠ ⊤ ** exact (Lp.memℒp _).snorm_ne_top ** case hb α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : ℝ≥0 f : { x // x ∈ Lp E p } g : { x // x ∈ Lp F p } h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊ this : snorm (fun x => ↑↑f x) p μ ≤ c • snorm (fun x => ↑↑g x) p μ ⊢ c • snorm (fun x => ↑↑g x) p μ ≠ ⊤ ** exact ENNReal.mul_ne_top ENNReal.coe_ne_top (Lp.memℒp _).snorm_ne_top ** Qed
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MeasureTheory.Lp.norm_le_norm_of_ae_le ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } g : { x // x ∈ Lp F p } h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ‖↑↑g x‖ ⊢ ‖f‖ ≤ ‖g‖ ** rw [norm_def, norm_def, ENNReal.toReal_le_toReal (snorm_ne_top _) (snorm_ne_top _)] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G f : { x // x ∈ Lp E p } g : { x // x ∈ Lp F p } h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ‖↑↑g x‖ ⊢ snorm (↑↑f) p μ ≤ snorm (↑↑g) p μ ** exact snorm_mono_ae h ** Qed
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MeasureTheory.Lp.nnnorm_le_of_ae_bound ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ≥0 hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C ⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹ * C ** by_cases hμ : μ = 0 ** case neg α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ≥0 hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C hμ : ¬μ = 0 ⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹ * C ** rw [← ENNReal.coe_le_coe, nnnorm_def, ENNReal.coe_toNNReal (snorm_ne_top _)] ** case neg α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ≥0 hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C hμ : ¬μ = 0 ⊢ snorm (↑↑f) p μ ≤ ↑(measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹ * C) ** refine' (snorm_le_of_ae_nnnorm_bound hfC).trans_eq _ ** case neg α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ≥0 hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C hμ : ¬μ = 0 ⊢ C • ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ = ↑(measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹ * C) ** rw [← coe_measureUnivNNReal μ, ENNReal.coe_rpow_of_ne_zero (measureUnivNNReal_pos hμ).ne',
ENNReal.coe_mul, mul_comm, ENNReal.smul_def, smul_eq_mul] ** case pos α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ≥0 hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C hμ : μ = 0 ⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹ * C ** by_cases hp : p.toReal⁻¹ = 0 ** case pos α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ≥0 hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C hμ : μ = 0 hp : (ENNReal.toReal p)⁻¹ = 0 ⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹ * C ** simp [hp, hμ, nnnorm_def] ** case neg α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ≥0 hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C hμ : μ = 0 hp : ¬(ENNReal.toReal p)⁻¹ = 0 ⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹ * C ** simp [hμ, nnnorm_def, Real.zero_rpow hp] ** Qed
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MeasureTheory.Lp.norm_le_of_ae_bound ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ hC : 0 ≤ C hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ C ⊢ ‖f‖ ≤ ↑(measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹) * C ** lift C to ℝ≥0 using hC ** case intro α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ≥0 hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ↑C ⊢ ‖f‖ ≤ ↑(measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹) * ↑C ** have := nnnorm_le_of_ae_bound hfC ** case intro α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝³ : NormedAddCommGroup E inst✝² : NormedAddCommGroup F inst✝¹ : NormedAddCommGroup G inst✝ : IsFiniteMeasure μ f : { x // x ∈ Lp E p } C : ℝ≥0 hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ↑C this : ‖f‖₊ ≤ measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹ * C ⊢ ‖f‖ ≤ ↑(measureUnivNNReal μ ^ (ENNReal.toReal p)⁻¹) * ↑C ** rwa [← NNReal.coe_le_coe, NNReal.coe_mul, NNReal.coe_rpow] at this ** Qed
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MeasureTheory.Lp.const_smul_mem_Lp ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedAddCommGroup G 𝕜 : Type u_5 𝕜' : Type u_6 inst✝⁵ : NormedRing 𝕜 inst✝⁴ : NormedRing 𝕜' inst✝³ : Module 𝕜 E inst✝² : Module 𝕜' E inst✝¹ : BoundedSMul 𝕜 E inst✝ : BoundedSMul 𝕜' E c : 𝕜 f : { x // x ∈ Lp E p } ⊢ c • ↑f ∈ Lp E p ** rw [mem_Lp_iff_snorm_lt_top, snorm_congr_ae (AEEqFun.coeFn_smul _ _)] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedAddCommGroup G 𝕜 : Type u_5 𝕜' : Type u_6 inst✝⁵ : NormedRing 𝕜 inst✝⁴ : NormedRing 𝕜' inst✝³ : Module 𝕜 E inst✝² : Module 𝕜' E inst✝¹ : BoundedSMul 𝕜 E inst✝ : BoundedSMul 𝕜' E c : 𝕜 f : { x // x ∈ Lp E p } ⊢ snorm (c • ↑↑f) p μ < ⊤ ** refine' (snorm_const_smul_le _ _).trans_lt _ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedAddCommGroup G 𝕜 : Type u_5 𝕜' : Type u_6 inst✝⁵ : NormedRing 𝕜 inst✝⁴ : NormedRing 𝕜' inst✝³ : Module 𝕜 E inst✝² : Module 𝕜' E inst✝¹ : BoundedSMul 𝕜 E inst✝ : BoundedSMul 𝕜' E c : 𝕜 f : { x // x ∈ Lp E p } ⊢ ‖c‖₊ • snorm (↑↑f) p μ < ⊤ ** rw [ENNReal.smul_def, smul_eq_mul, ENNReal.mul_lt_top_iff] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝⁸ : NormedAddCommGroup E inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedAddCommGroup G 𝕜 : Type u_5 𝕜' : Type u_6 inst✝⁵ : NormedRing 𝕜 inst✝⁴ : NormedRing 𝕜' inst✝³ : Module 𝕜 E inst✝² : Module 𝕜' E inst✝¹ : BoundedSMul 𝕜 E inst✝ : BoundedSMul 𝕜' E c : 𝕜 f : { x // x ∈ Lp E p } ⊢ ↑‖c‖₊ < ⊤ ∧ snorm (↑↑f) p μ < ⊤ ∨ ↑‖c‖₊ = 0 ∨ snorm (↑↑f) p μ = 0 ** exact Or.inl ⟨ENNReal.coe_lt_top, f.prop⟩ ** Qed
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MeasureTheory.snormEssSup_indicator_le ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → G ⊢ snormEssSup (Set.indicator s f) μ ≤ snormEssSup f μ ** refine' essSup_mono_ae (eventually_of_forall fun x => _) ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → G x : α ⊢ (fun x => ↑‖Set.indicator s f x‖₊) x ≤ (fun x => ↑‖f x‖₊) x ** rw [ENNReal.coe_le_coe, nnnorm_indicator_eq_indicator_nnnorm] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → G x : α ⊢ Set.indicator s (fun a => ‖f a‖₊) x ≤ ‖f x‖₊ ** exact Set.indicator_le_self s _ x ** Qed
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MeasureTheory.snormEssSup_indicator_const_eq ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hμs : ↑↑μ s ≠ 0 ⊢ snormEssSup (Set.indicator s fun x => c) μ = ↑‖c‖₊ ** refine' le_antisymm (snormEssSup_indicator_const_le s c) _ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hμs : ↑↑μ s ≠ 0 ⊢ ↑‖c‖₊ ≤ snormEssSup (Set.indicator s fun x => c) μ ** by_contra' h ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hμs : ↑↑μ s ≠ 0 h : snormEssSup (Set.indicator s fun x => c) μ < ↑‖c‖₊ ⊢ False ** have h' := ae_iff.mp (ae_lt_of_essSup_lt h) ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hμs : ↑↑μ s ≠ 0 h : snormEssSup (Set.indicator s fun x => c) μ < ↑‖c‖₊ h' : ↑↑μ {a | ¬↑‖Set.indicator s (fun x => c) a‖₊ < ↑‖c‖₊} = 0 ⊢ False ** push_neg at h' ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hμs : ↑↑μ s ≠ 0 h : snormEssSup (Set.indicator s fun x => c) μ < ↑‖c‖₊ h' : ↑↑μ {a | ↑‖c‖₊ ≤ ↑‖Set.indicator s (fun x => c) a‖₊} = 0 ⊢ False ** refine' hμs (measure_mono_null (fun x hx_mem => _) h') ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hμs : ↑↑μ s ≠ 0 h : snormEssSup (Set.indicator s fun x => c) μ < ↑‖c‖₊ h' : ↑↑μ {a | ↑‖c‖₊ ≤ ↑‖Set.indicator s (fun x => c) a‖₊} = 0 x : α hx_mem : x ∈ s ⊢ x ∈ {a | ↑‖c‖₊ ≤ ↑‖Set.indicator s (fun x => c) a‖₊} ** rw [Set.mem_setOf_eq, Set.indicator_of_mem hx_mem] ** Qed
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MeasureTheory.snorm_indicator_le ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ f : α → E s : Set α ⊢ snorm (Set.indicator s f) p μ ≤ snorm f p μ ** refine' snorm_mono_ae (eventually_of_forall fun x => _) ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ f : α → E s : Set α x : α ⊢ ‖Set.indicator s f x‖ ≤ ‖f x‖ ** suffices ‖s.indicator f x‖₊ ≤ ‖f x‖₊ by exact NNReal.coe_mono this ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ f : α → E s : Set α x : α ⊢ ‖Set.indicator s f x‖₊ ≤ ‖f x‖₊ ** rw [nnnorm_indicator_eq_indicator_nnnorm] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ f : α → E s : Set α x : α ⊢ Set.indicator s (fun a => ‖f a‖₊) x ≤ ‖f x‖₊ ** exact s.indicator_le_self _ x ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ f : α → E s : Set α x : α this : ‖Set.indicator s f x‖₊ ≤ ‖f x‖₊ ⊢ ‖Set.indicator s f x‖ ≤ ‖f x‖ ** exact NNReal.coe_mono this ** Qed
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MeasureTheory.snorm_indicator_const₀ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hs : NullMeasurableSet s hp : p ≠ 0 hp_top : p ≠ ⊤ hp_pos : 0 < ENNReal.toReal p ⊢ (∫⁻ (x : α), ↑‖Set.indicator s (fun x => c) x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) = (∫⁻ (x : α), Set.indicator s (fun x => ↑‖c‖₊ ^ ENNReal.toReal p) x ∂μ) ^ (1 / ENNReal.toReal p) ** congr 2 ** case e_a.e_f α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hs : NullMeasurableSet s hp : p ≠ 0 hp_top : p ≠ ⊤ hp_pos : 0 < ENNReal.toReal p ⊢ (fun x => ↑‖Set.indicator s (fun x => c) x‖₊ ^ ENNReal.toReal p) = fun x => Set.indicator s (fun x => ↑‖c‖₊ ^ ENNReal.toReal p) x ** refine (Set.comp_indicator_const c (fun x : G ↦ (‖x‖₊ : ℝ≥0∞) ^ p.toReal) ?_) ** case e_a.e_f α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hs : NullMeasurableSet s hp : p ≠ 0 hp_top : p ≠ ⊤ hp_pos : 0 < ENNReal.toReal p ⊢ (fun x => ↑‖x‖₊ ^ ENNReal.toReal p) 0 = 0 ** simp [hp_pos] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hs : NullMeasurableSet s hp : p ≠ 0 hp_top : p ≠ ⊤ hp_pos : 0 < ENNReal.toReal p ⊢ (∫⁻ (x : α), Set.indicator s (fun x => ↑‖c‖₊ ^ ENNReal.toReal p) x ∂μ) ^ (1 / ENNReal.toReal p) = ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p) ** rw [lintegral_indicator_const₀ hs, ENNReal.mul_rpow_of_nonneg, ← ENNReal.rpow_mul,
mul_one_div_cancel hp_pos.ne', ENNReal.rpow_one] ** case hz α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hs : NullMeasurableSet s hp : p ≠ 0 hp_top : p ≠ ⊤ hp_pos : 0 < ENNReal.toReal p ⊢ 0 ≤ 1 / ENNReal.toReal p ** positivity ** Qed
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MeasureTheory.snorm_indicator_const_le ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G p : ℝ≥0∞ ⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p) ** rcases eq_or_ne p 0 with (rfl | hp) ** case inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G p : ℝ≥0∞ hp : p ≠ 0 ⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p) ** rcases eq_or_ne p ∞ with (rfl | h'p) ** case inr.inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G p : ℝ≥0∞ hp : p ≠ 0 h'p : p ≠ ⊤ ⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p) ** let t := toMeasurable μ s ** case inr.inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G p : ℝ≥0∞ hp : p ≠ 0 h'p : p ≠ ⊤ t : Set α := toMeasurable μ s ⊢ snorm (Set.indicator s fun x => c) p μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p) ** calc
snorm (s.indicator fun _ => c) p μ ≤ snorm (t.indicator fun _ => c) p μ :=
snorm_mono (norm_indicator_le_of_subset (subset_toMeasurable _ _) _)
_ = ‖c‖₊ * μ t ^ (1 / p.toReal) :=
(snorm_indicator_const (measurableSet_toMeasurable _ _) hp h'p)
_ = ‖c‖₊ * μ s ^ (1 / p.toReal) := by rw [measure_toMeasurable] ** case inl α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G ⊢ snorm (Set.indicator s fun x => c) 0 μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal 0) ** simp only [snorm_exponent_zero, zero_le'] ** case inr.inl α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hp : ⊤ ≠ 0 ⊢ snorm (Set.indicator s fun x => c) ⊤ μ ≤ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal ⊤) ** simp only [snorm_exponent_top, ENNReal.top_toReal, _root_.div_zero, ENNReal.rpow_zero, mul_one] ** case inr.inl α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G hp : ⊤ ≠ 0 ⊢ snormEssSup (Set.indicator s fun x => c) μ ≤ ↑‖c‖₊ ** exact snormEssSup_indicator_const_le _ _ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α c : G p : ℝ≥0∞ hp : p ≠ 0 h'p : p ≠ ⊤ t : Set α := toMeasurable μ s ⊢ ↑‖c‖₊ * ↑↑μ t ^ (1 / ENNReal.toReal p) = ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p) ** rw [measure_toMeasurable] ** Qed
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MeasureTheory.snorm_indicator_eq_snorm_restrict ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s ⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s) ** by_cases hp_zero : p = 0 ** case neg α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 ⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s) ** by_cases hp_top : p = ∞ ** case neg α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : ¬p = ⊤ ⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s) ** simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_top] ** case neg α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : ¬p = ⊤ ⊢ (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) = (∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) ** suffices (∫⁻ x, (‖s.indicator f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) =
∫⁻ x in s, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ by rw [this] ** case neg α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : ¬p = ⊤ ⊢ ∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ = ∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ ** rw [← lintegral_indicator _ hs] ** case neg α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : ¬p = ⊤ ⊢ ∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ = ∫⁻ (a : α), Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a ∂μ ** congr ** case neg.e_f α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : ¬p = ⊤ ⊢ (fun x => ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p) = fun a => Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a ** simp_rw [nnnorm_indicator_eq_indicator_nnnorm, ENNReal.coe_indicator] ** case neg.e_f α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : ¬p = ⊤ ⊢ (fun x => Set.indicator s (fun x => ↑‖f x‖₊) x ^ ENNReal.toReal p) = fun a => Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a ** have h_zero : (fun x => x ^ p.toReal) (0 : ℝ≥0∞) = 0 := by
simp [ENNReal.toReal_pos hp_zero hp_top] ** case neg.e_f α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : ¬p = ⊤ h_zero : (fun x => x ^ ENNReal.toReal p) 0 = 0 ⊢ (fun x => Set.indicator s (fun x => ↑‖f x‖₊) x ^ ENNReal.toReal p) = fun a => Set.indicator s (fun x => ↑‖f x‖₊ ^ ENNReal.toReal p) a ** exact (Set.indicator_comp_of_zero (g := fun x : ℝ≥0∞ => x ^ p.toReal) h_zero).symm ** case pos α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : p = 0 ⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s) ** simp only [hp_zero, snorm_exponent_zero] ** case pos α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : p = ⊤ ⊢ snorm (Set.indicator s f) p μ = snorm f p (Measure.restrict μ s) ** simp_rw [hp_top, snorm_exponent_top] ** case pos α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : p = ⊤ ⊢ snormEssSup (Set.indicator s f) μ = snormEssSup f (Measure.restrict μ s) ** exact snormEssSup_indicator_eq_snormEssSup_restrict hs ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : ¬p = ⊤ this : ∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ = ∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ ⊢ (∫⁻ (x : α), ↑‖Set.indicator s f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) = (∫⁻ (x : α) in s, ↑‖f x‖₊ ^ ENNReal.toReal p ∂μ) ^ (1 / ENNReal.toReal p) ** rw [this] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf : AEStronglyMeasurable f✝ μ s : Set α f : α → F hs : MeasurableSet s hp_zero : ¬p = 0 hp_top : ¬p = ⊤ ⊢ (fun x => x ^ ENNReal.toReal p) 0 = 0 ** simp [ENNReal.toReal_pos hp_zero hp_top] ** Qed
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MeasureTheory.memℒp_indicator_iff_restrict ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f : α → E hf : AEStronglyMeasurable f μ s : Set α hs : MeasurableSet s ⊢ Memℒp (Set.indicator s f) p ↔ Memℒp f p ** simp [Memℒp, aestronglyMeasurable_indicator_iff hs, snorm_indicator_eq_snorm_restrict hs] ** Qed
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MeasureTheory.Memℒp.memℒp_of_exponent_le_of_measure_support_ne_top ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q✝ : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf✝ : AEStronglyMeasurable f✝ μ p q : ℝ≥0∞ f : α → E hfq : Memℒp f q s : Set α hf : ∀ (x : α), ¬x ∈ s → f x = 0 hs : ↑↑μ s ≠ ⊤ hpq : p ≤ q ⊢ Memℒp f p ** have : (toMeasurable μ s).indicator f = f := by
apply Set.indicator_eq_self.2
apply Function.support_subset_iff'.2 (fun x hx ↦ hf x ?_)
contrapose! hx
exact subset_toMeasurable μ s hx ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q✝ : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf✝ : AEStronglyMeasurable f✝ μ p q : ℝ≥0∞ f : α → E hfq : Memℒp f q s : Set α hf : ∀ (x : α), ¬x ∈ s → f x = 0 hs : ↑↑μ s ≠ ⊤ hpq : p ≤ q this : Set.indicator (toMeasurable μ s) f = f ⊢ Memℒp f p ** rw [← this, memℒp_indicator_iff_restrict (measurableSet_toMeasurable μ s)] at hfq ⊢ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q✝ : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf✝ : AEStronglyMeasurable f✝ μ p q : ℝ≥0∞ f : α → E s : Set α hfq : Memℒp f q hf : ∀ (x : α), ¬x ∈ s → f x = 0 hs : ↑↑μ s ≠ ⊤ hpq : p ≤ q this : Set.indicator (toMeasurable μ s) f = f ⊢ Memℒp f p ** have : Fact (μ (toMeasurable μ s) < ∞) := ⟨by simpa [lt_top_iff_ne_top] using hs⟩ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q✝ : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf✝ : AEStronglyMeasurable f✝ μ p q : ℝ≥0∞ f : α → E s : Set α hfq : Memℒp f q hf : ∀ (x : α), ¬x ∈ s → f x = 0 hs : ↑↑μ s ≠ ⊤ hpq : p ≤ q this✝ : Set.indicator (toMeasurable μ s) f = f this : Fact (↑↑μ (toMeasurable μ s) < ⊤) ⊢ Memℒp f p ** exact memℒp_of_exponent_le hfq hpq ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q✝ : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf✝ : AEStronglyMeasurable f✝ μ p q : ℝ≥0∞ f : α → E hfq : Memℒp f q s : Set α hf : ∀ (x : α), ¬x ∈ s → f x = 0 hs : ↑↑μ s ≠ ⊤ hpq : p ≤ q ⊢ Set.indicator (toMeasurable μ s) f = f ** apply Set.indicator_eq_self.2 ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q✝ : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf✝ : AEStronglyMeasurable f✝ μ p q : ℝ≥0∞ f : α → E hfq : Memℒp f q s : Set α hf : ∀ (x : α), ¬x ∈ s → f x = 0 hs : ↑↑μ s ≠ ⊤ hpq : p ≤ q ⊢ Function.support f ⊆ toMeasurable μ s ** apply Function.support_subset_iff'.2 (fun x hx ↦ hf x ?_) ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q✝ : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf✝ : AEStronglyMeasurable f✝ μ p q : ℝ≥0∞ f : α → E hfq : Memℒp f q s : Set α hf : ∀ (x : α), ¬x ∈ s → f x = 0 hs : ↑↑μ s ≠ ⊤ hpq : p ≤ q x : α hx : ¬x ∈ toMeasurable μ s ⊢ ¬x ∈ s ** contrapose! hx ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q✝ : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf✝ : AEStronglyMeasurable f✝ μ p q : ℝ≥0∞ f : α → E hfq : Memℒp f q s : Set α hf : ∀ (x : α), ¬x ∈ s → f x = 0 hs : ↑↑μ s ≠ ⊤ hpq : p ≤ q x : α hx : x ∈ s ⊢ x ∈ toMeasurable μ s ** exact subset_toMeasurable μ s hx ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q✝ : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f✝ : α → E hf✝ : AEStronglyMeasurable f✝ μ p q : ℝ≥0∞ f : α → E s : Set α hfq : Memℒp f q hf : ∀ (x : α), ¬x ∈ s → f x = 0 hs : ↑↑μ s ≠ ⊤ hpq : p ≤ q this : Set.indicator (toMeasurable μ s) f = f ⊢ ↑↑μ (toMeasurable μ s) < ⊤ ** simpa [lt_top_iff_ne_top] using hs ** Qed
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MeasureTheory.memℒp_indicator_const ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α p : ℝ≥0∞ hs : MeasurableSet s c : E hμsc : c = 0 ∨ ↑↑μ s ≠ ⊤ ⊢ Memℒp (Set.indicator s fun x => c) p ** rw [memℒp_indicator_iff_restrict hs] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α p : ℝ≥0∞ hs : MeasurableSet s c : E hμsc : c = 0 ∨ ↑↑μ s ≠ ⊤ ⊢ Memℒp (fun x => c) p ** rcases hμsc with rfl | hμ ** case inl α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c : E f : α → E hf : AEStronglyMeasurable f μ s : Set α p : ℝ≥0∞ hs : MeasurableSet s ⊢ Memℒp (fun x => 0) p ** exact zero_memℒp ** case inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α p : ℝ≥0∞ hs : MeasurableSet s c : E hμ : ↑↑μ s ≠ ⊤ ⊢ Memℒp (fun x => c) p ** have := Fact.mk hμ.lt_top ** case inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p✝ : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ s : Set α p : ℝ≥0∞ hs : MeasurableSet s c : E hμ : ↑↑μ s ≠ ⊤ this : Fact (↑↑μ s < ⊤) ⊢ Memℒp (fun x => c) p ** apply memℒp_const ** Qed
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MeasureTheory.exists_snorm_indicator_le ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 ⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun x => c) p μ ≤ ε ** rcases eq_or_ne p 0 with (rfl | h'p) ** case inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 ⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun x => c) p μ ≤ ε ** have hp₀ : 0 < p := bot_lt_iff_ne_bot.2 h'p ** case inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p ⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun x => c) p μ ≤ ε ** have hp₀' : 0 ≤ 1 / p.toReal := div_nonneg zero_le_one ENNReal.toReal_nonneg ** case inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p ⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun x => c) p μ ≤ ε ** have hp₀'' : 0 < p.toReal := ENNReal.toReal_pos hp₀.ne' hp ** case inr α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p ⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun x => c) p μ ≤ ε ** obtain ⟨η, hη_pos, hη_le⟩ :
∃ η : ℝ≥0, 0 < η ∧ (‖c‖₊ : ℝ≥0∞) * (η : ℝ≥0∞) ^ (1 / p.toReal) ≤ ε := by
have :
Filter.Tendsto (fun x : ℝ≥0 => ((‖c‖₊ * x ^ (1 / p.toReal) : ℝ≥0) : ℝ≥0∞)) (𝓝 0)
(𝓝 (0 : ℝ≥0)) := by
rw [ENNReal.tendsto_coe]
convert (NNReal.continuousAt_rpow_const (Or.inr hp₀')).tendsto.const_mul _
simp [hp₀''.ne']
have hε' : 0 < ε := hε.bot_lt
obtain ⟨δ, hδ, hδε'⟩ :=
NNReal.nhds_zero_basis.eventually_iff.mp (eventually_le_of_tendsto_lt hε' this)
obtain ⟨η, hη, hηδ⟩ := exists_between hδ
refine' ⟨η, hη, _⟩
rw [ENNReal.coe_rpow_of_nonneg _ hp₀', ← ENNReal.coe_mul]
exact hδε' hηδ ** case inr.intro.intro α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p η : ℝ≥0 hη_pos : 0 < η hη_le : ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ≤ ε ⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun x => c) p μ ≤ ε ** refine' ⟨η, hη_pos, fun s hs => _⟩ ** case inr.intro.intro α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p η : ℝ≥0 hη_pos : 0 < η hη_le : ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ≤ ε s : Set α hs : ↑↑μ s ≤ ↑η ⊢ snorm (Set.indicator s fun x => c) p μ ≤ ε ** refine' (snorm_indicator_const_le _ _).trans (le_trans _ hη_le) ** case inr.intro.intro α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p η : ℝ≥0 hη_pos : 0 < η hη_le : ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ≤ ε s : Set α hs : ↑↑μ s ≤ ↑η ⊢ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p) ≤ ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ** exact mul_le_mul_left' (ENNReal.rpow_le_rpow hs hp₀') _ ** case inl α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ c : E ε : ℝ≥0∞ hε : ε ≠ 0 hp : 0 ≠ ⊤ ⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun x => c) 0 μ ≤ ε ** exact ⟨1, zero_lt_one, fun s _ => by simp⟩ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ c : E ε : ℝ≥0∞ hε : ε ≠ 0 hp : 0 ≠ ⊤ s : Set α x✝ : ↑↑μ s ≤ ↑1 ⊢ snorm (Set.indicator s fun x => c) 0 μ ≤ ε ** simp ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p ⊢ ∃ η, 0 < η ∧ ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ≤ ε ** have :
Filter.Tendsto (fun x : ℝ≥0 => ((‖c‖₊ * x ^ (1 / p.toReal) : ℝ≥0) : ℝ≥0∞)) (𝓝 0)
(𝓝 (0 : ℝ≥0)) := by
rw [ENNReal.tendsto_coe]
convert (NNReal.continuousAt_rpow_const (Or.inr hp₀')).tendsto.const_mul _
simp [hp₀''.ne'] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p this : Tendsto (fun x => ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p))) (𝓝 0) (𝓝 ↑0) ⊢ ∃ η, 0 < η ∧ ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ≤ ε ** have hε' : 0 < ε := hε.bot_lt ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p this : Tendsto (fun x => ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p))) (𝓝 0) (𝓝 ↑0) hε' : 0 < ε ⊢ ∃ η, 0 < η ∧ ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ≤ ε ** obtain ⟨δ, hδ, hδε'⟩ :=
NNReal.nhds_zero_basis.eventually_iff.mp (eventually_le_of_tendsto_lt hε' this) ** case intro.intro α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p this : Tendsto (fun x => ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p))) (𝓝 0) (𝓝 ↑0) hε' : 0 < ε δ : ℝ≥0 hδ : 0 < δ hδε' : ∀ ⦃x : ℝ≥0⦄, x ∈ Set.Iio δ → ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p)) ≤ ε ⊢ ∃ η, 0 < η ∧ ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ≤ ε ** obtain ⟨η, hη, hηδ⟩ := exists_between hδ ** case intro.intro.intro.intro α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p this : Tendsto (fun x => ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p))) (𝓝 0) (𝓝 ↑0) hε' : 0 < ε δ : ℝ≥0 hδ : 0 < δ hδε' : ∀ ⦃x : ℝ≥0⦄, x ∈ Set.Iio δ → ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p)) ≤ ε η : ℝ≥0 hη : 0 < η hηδ : η < δ ⊢ ∃ η, 0 < η ∧ ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ≤ ε ** refine' ⟨η, hη, _⟩ ** case intro.intro.intro.intro α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p this : Tendsto (fun x => ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p))) (𝓝 0) (𝓝 ↑0) hε' : 0 < ε δ : ℝ≥0 hδ : 0 < δ hδε' : ∀ ⦃x : ℝ≥0⦄, x ∈ Set.Iio δ → ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p)) ≤ ε η : ℝ≥0 hη : 0 < η hηδ : η < δ ⊢ ↑‖c‖₊ * ↑η ^ (1 / ENNReal.toReal p) ≤ ε ** rw [ENNReal.coe_rpow_of_nonneg _ hp₀', ← ENNReal.coe_mul] ** case intro.intro.intro.intro α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p this : Tendsto (fun x => ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p))) (𝓝 0) (𝓝 ↑0) hε' : 0 < ε δ : ℝ≥0 hδ : 0 < δ hδε' : ∀ ⦃x : ℝ≥0⦄, x ∈ Set.Iio δ → ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p)) ≤ ε η : ℝ≥0 hη : 0 < η hηδ : η < δ ⊢ ↑(‖c‖₊ * η ^ (1 / ENNReal.toReal p)) ≤ ε ** exact hδε' hηδ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p ⊢ Tendsto (fun x => ↑(‖c‖₊ * x ^ (1 / ENNReal.toReal p))) (𝓝 0) (𝓝 ↑0) ** rw [ENNReal.tendsto_coe] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p ⊢ Tendsto (fun x => ‖c‖₊ * x ^ (1 / ENNReal.toReal p)) (𝓝 0) (𝓝 0) ** convert (NNReal.continuousAt_rpow_const (Or.inr hp₀')).tendsto.const_mul _ ** case h.e'_5.h.e'_3 α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G c✝ : E f : α → E hf : AEStronglyMeasurable f μ hp : p ≠ ⊤ c : E ε : ℝ≥0∞ hε : ε ≠ 0 h'p : p ≠ 0 hp₀ : 0 < p hp₀' : 0 ≤ 1 / ENNReal.toReal p hp₀'' : 0 < ENNReal.toReal p ⊢ 0 = ‖c‖₊ * 0 ^ (1 / ENNReal.toReal p) ** simp [hp₀''.ne'] ** Qed
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MeasureTheory.norm_indicatorConstLp ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E hp_ne_zero : p ≠ 0 hp_ne_top : p ≠ ⊤ ⊢ ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * ENNReal.toReal (↑↑μ s) ^ (1 / ENNReal.toReal p) ** rw [Lp.norm_def, snorm_congr_ae indicatorConstLp_coeFn,
snorm_indicator_const hs hp_ne_zero hp_ne_top, ENNReal.toReal_mul, ENNReal.toReal_rpow,
ENNReal.coe_toReal, coe_nnnorm] ** Qed
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MeasureTheory.norm_indicatorConstLp_le ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ ‖indicatorConstLp p hs hμs c‖ ≤ ‖c‖ * ENNReal.toReal (↑↑μ s) ^ (1 / ENNReal.toReal p) ** rw [indicatorConstLp, Lp.norm_toLp] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ ENNReal.toReal (snorm (indicator s fun x => c) p μ) ≤ ‖c‖ * ENNReal.toReal (↑↑μ s) ^ (1 / ENNReal.toReal p) ** refine ENNReal.toReal_le_of_le_ofReal (by positivity) ?_ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ snorm (indicator s fun x => c) p μ ≤ ENNReal.ofReal (‖c‖ * ENNReal.toReal (↑↑μ s) ^ (1 / ENNReal.toReal p)) ** refine (snorm_indicator_const_le _ _).trans_eq ?_ ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p) = ENNReal.ofReal (‖c‖ * ENNReal.toReal (↑↑μ s) ^ (1 / ENNReal.toReal p)) ** rw [← coe_nnnorm, ENNReal.ofReal_mul (NNReal.coe_nonneg _), ENNReal.ofReal_coe_nnreal,
ENNReal.toReal_rpow, ENNReal.ofReal_toReal] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ ↑↑μ s ^ (1 / ENNReal.toReal p) ≠ ⊤ ** exact ENNReal.rpow_ne_top_of_nonneg (by positivity) hμs ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ 0 ≤ ‖c‖ * ENNReal.toReal (↑↑μ s) ^ (1 / ENNReal.toReal p) ** positivity ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ 0 ≤ 1 / ENNReal.toReal p ** positivity ** Qed
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MeasureTheory.indicatorConstLp_empty ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ ↑↑μ ∅ ≠ ⊤ ** simp ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ indicatorConstLp p (_ : MeasurableSet ∅) (_ : ↑↑μ ∅ ≠ ⊤) c = 0 ** rw [Lp.eq_zero_iff_ae_eq_zero] ** α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ ↑↑(indicatorConstLp p (_ : MeasurableSet ∅) (_ : ↑↑μ ∅ ≠ ⊤) c) =ᵐ[μ] 0 ** convert indicatorConstLp_coeFn (E := E) ** case h.e'_5 α : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace α p : ℝ≥0∞ q : ℝ μ ν : Measure α inst✝² : NormedAddCommGroup E inst✝¹ : NormedAddCommGroup F inst✝ : NormedAddCommGroup G s : Set α hs : MeasurableSet s hμs : ↑↑μ s ≠ ⊤ c : E ⊢ 0 = indicator ∅ fun x => c ** simp [Set.indicator_empty', Pi.zero_def] ** Qed
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