formal
stringlengths 41
427k
| informal
stringclasses 1
value |
|---|---|
MeasureTheory.set_lintegral_univ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ ⊢ ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ ** rw [Measure.restrict_univ] ** Qed
| |
MeasureTheory.set_lintegral_measure_zero ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α f : α → ℝ≥0∞ hs' : ↑↑μ s = 0 ⊢ ∫⁻ (x : α) in s, f x ∂μ = 0 ** convert lintegral_zero_measure _ ** case h.e'_2.h.e'_3.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α f : α → ℝ≥0∞ hs' : ↑↑μ s = 0 ⊢ Measure.restrict μ s = 0 ** exact Measure.restrict_eq_zero.2 hs' ** Qed
| |
MeasureTheory.lintegral_finset_sum' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Finset β f : β → α → ℝ≥0∞ hf : ∀ (b : β), b ∈ s → AEMeasurable (f b) ⊢ ∫⁻ (a : α), ∑ b in s, f b a ∂μ = ∑ b in s, ∫⁻ (a : α), f b a ∂μ ** induction' s using Finset.induction_on with a s has ih ** case empty α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Finset β f : β → α → ℝ≥0∞ hf✝ : ∀ (b : β), b ∈ s → AEMeasurable (f b) hf : ∀ (b : β), b ∈ ∅ → AEMeasurable (f b) ⊢ ∫⁻ (a : α), ∑ b in ∅, f b a ∂μ = ∑ b in ∅, ∫⁻ (a : α), f b a ∂μ ** simp ** case insert α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s✝ : Finset β f : β → α → ℝ≥0∞ hf✝ : ∀ (b : β), b ∈ s✝ → AEMeasurable (f b) a : β s : Finset β has : ¬a ∈ s ih : (∀ (b : β), b ∈ s → AEMeasurable (f b)) → ∫⁻ (a : α), ∑ b in s, f b a ∂μ = ∑ b in s, ∫⁻ (a : α), f b a ∂μ hf : ∀ (b : β), b ∈ insert a s → AEMeasurable (f b) ⊢ ∫⁻ (a_1 : α), ∑ b in insert a s, f b a_1 ∂μ = ∑ b in insert a s, ∫⁻ (a : α), f b a ∂μ ** simp only [Finset.sum_insert has] ** case insert α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s✝ : Finset β f : β → α → ℝ≥0∞ hf✝ : ∀ (b : β), b ∈ s✝ → AEMeasurable (f b) a : β s : Finset β has : ¬a ∈ s ih : (∀ (b : β), b ∈ s → AEMeasurable (f b)) → ∫⁻ (a : α), ∑ b in s, f b a ∂μ = ∑ b in s, ∫⁻ (a : α), f b a ∂μ hf : ∀ (b : β), b ∈ insert a s → AEMeasurable (f b) ⊢ ∫⁻ (a_1 : α), f a a_1 + ∑ b in s, f b a_1 ∂μ = ∫⁻ (a_1 : α), f a a_1 ∂μ + ∑ b in s, ∫⁻ (a : α), f b a ∂μ ** rw [Finset.forall_mem_insert] at hf ** case insert α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s✝ : Finset β f : β → α → ℝ≥0∞ hf✝ : ∀ (b : β), b ∈ s✝ → AEMeasurable (f b) a : β s : Finset β has : ¬a ∈ s ih : (∀ (b : β), b ∈ s → AEMeasurable (f b)) → ∫⁻ (a : α), ∑ b in s, f b a ∂μ = ∑ b in s, ∫⁻ (a : α), f b a ∂μ hf : AEMeasurable (f a) ∧ ∀ (x : β), x ∈ s → AEMeasurable (f x) ⊢ ∫⁻ (a_1 : α), f a a_1 + ∑ b in s, f b a_1 ∂μ = ∫⁻ (a_1 : α), f a a_1 ∂μ + ∑ b in s, ∫⁻ (a : α), f b a ∂μ ** rw [lintegral_add_left' hf.1, ih hf.2] ** Qed
| |
MeasureTheory.lintegral_const_mul ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), r * f a ∂μ = ∫⁻ (a : α), ⨆ n, ↑(const α r * eapprox f n) a ∂μ ** congr ** case e_f α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ (fun a => r * f a) = fun a => ⨆ n, ↑(const α r * eapprox f n) a ** funext a ** case e_f.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f a : α ⊢ r * f a = ⨆ n, ↑(const α r * eapprox f n) a ** rw [← iSup_eapprox_apply f hf, ENNReal.mul_iSup] ** case e_f.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f a : α ⊢ ⨆ i, r * ↑(eapprox f i) a = ⨆ n, ↑(const α r * eapprox f n) a ** rfl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), ⨆ n, ↑(const α r * eapprox f n) a ∂μ = ⨆ n, r * SimpleFunc.lintegral (eapprox f n) μ ** rw [lintegral_iSup] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ ⨆ n, ∫⁻ (a : α), ↑(const α r * eapprox f n) a ∂μ = ⨆ n, r * SimpleFunc.lintegral (eapprox f n) μ ** congr ** case e_s α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ (fun n => ∫⁻ (a : α), ↑(const α r * eapprox f n) a ∂μ) = fun n => r * SimpleFunc.lintegral (eapprox f n) μ ** funext n ** case e_s.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f n : ℕ ⊢ ∫⁻ (a : α), ↑(const α r * eapprox f n) a ∂μ = r * SimpleFunc.lintegral (eapprox f n) μ ** rw [← SimpleFunc.const_mul_lintegral, ← SimpleFunc.lintegral_eq_lintegral] ** case hf α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ ∀ (n : ℕ), Measurable fun a => ↑(const α r * eapprox f n) a ** intro n ** case hf α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f n : ℕ ⊢ Measurable fun a => ↑(const α r * eapprox f n) a ** exact SimpleFunc.measurable _ ** case h_mono α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ Monotone fun n a => ↑(const α r * eapprox f n) a ** intro i j h a ** case h_mono α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f i j : ℕ h : i ≤ j a : α ⊢ (fun n a => ↑(const α r * eapprox f n) a) i a ≤ (fun n a => ↑(const α r * eapprox f n) a) j a ** exact mul_le_mul_left' (monotone_eapprox _ h _) _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ ⨆ n, r * SimpleFunc.lintegral (eapprox f n) μ = r * ∫⁻ (a : α), f a ∂μ ** rw [← ENNReal.mul_iSup, lintegral_eq_iSup_eapprox_lintegral hf] ** Qed
| |
MeasureTheory.lintegral_const_mul'' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : AEMeasurable f ⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ ** have A : ∫⁻ a, f a ∂μ = ∫⁻ a, hf.mk f a ∂μ := lintegral_congr_ae hf.ae_eq_mk ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ ⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ ** have B : ∫⁻ a, r * f a ∂μ = ∫⁻ a, r * hf.mk f a ∂μ :=
lintegral_congr_ae (EventuallyEq.fun_comp hf.ae_eq_mk _) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : AEMeasurable f A : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ B : ∫⁻ (a : α), r * f a ∂μ = ∫⁻ (a : α), r * AEMeasurable.mk f hf a ∂μ ⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ ** rw [A, B, lintegral_const_mul _ hf.measurable_mk] ** Qed
| |
MeasureTheory.lintegral_const_mul_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ ⊢ r * ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), r * f a ∂μ ** rw [lintegral, ENNReal.mul_iSup] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ ⊢ ⨆ i, r * ⨆ (_ : ↑i ≤ fun a => f a), SimpleFunc.lintegral i μ ≤ ∫⁻ (a : α), r * f a ∂μ ** refine' iSup_le fun s => _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ s : α →ₛ ℝ≥0∞ ⊢ r * ⨆ (_ : ↑s ≤ fun a => f a), SimpleFunc.lintegral s μ ≤ ∫⁻ (a : α), r * f a ∂μ ** rw [ENNReal.mul_iSup] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ s : α →ₛ ℝ≥0∞ ⊢ ⨆ (_ : ↑s ≤ fun a => f a), r * SimpleFunc.lintegral s μ ≤ ∫⁻ (a : α), r * f a ∂μ ** simp only [iSup_le_iff] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ s : α →ₛ ℝ≥0∞ ⊢ (↑s ≤ fun a => f a) → r * SimpleFunc.lintegral s μ ≤ ∫⁻ (a : α), r * f a ∂μ ** intro hs ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ s : α →ₛ ℝ≥0∞ hs : ↑s ≤ fun a => f a ⊢ r * SimpleFunc.lintegral s μ ≤ ∫⁻ (a : α), r * f a ∂μ ** rw [← SimpleFunc.const_mul_lintegral, lintegral] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ s : α →ₛ ℝ≥0∞ hs : ↑s ≤ fun a => f a ⊢ SimpleFunc.lintegral (const α r * s) μ ≤ ⨆ g, ⨆ (_ : ↑g ≤ fun a => r * f a), SimpleFunc.lintegral g μ ** refine' le_iSup_of_le (const α r * s) (le_iSup_of_le (fun x => _) le_rfl) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ s : α →ₛ ℝ≥0∞ hs : ↑s ≤ fun a => f a x : α ⊢ ↑(const α r * s) x ≤ (fun a => r * f a) x ** exact mul_le_mul_left' (hs x) _ ** Qed
| |
MeasureTheory.lintegral_const_mul' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ ⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ ** by_cases h : r = 0 ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ h : ¬r = 0 ⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ ** apply le_antisymm _ (lintegral_const_mul_le r f) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ h : ¬r = 0 ⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ ** have rinv : r * r⁻¹ = 1 := ENNReal.mul_inv_cancel h hr ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ h : ¬r = 0 rinv : r * r⁻¹ = 1 ⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ ** have rinv' : r⁻¹ * r = 1 := by
rw [mul_comm]
exact rinv ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ h : ¬r = 0 rinv : r * r⁻¹ = 1 rinv' : r⁻¹ * r = 1 ⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ ** have := lintegral_const_mul_le (μ := μ) r⁻¹ fun x => r * f x ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ h : ¬r = 0 rinv : r * r⁻¹ = 1 rinv' : r⁻¹ * r = 1 this : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), r⁻¹ * (r * f a) ∂μ ⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ ** simp [(mul_assoc _ _ _).symm, rinv'] at this ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ h : ¬r = 0 rinv : r * r⁻¹ = 1 rinv' : r⁻¹ * r = 1 this : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), f a ∂μ ⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ ** simpa [(mul_assoc _ _ _).symm, rinv] using mul_le_mul_left' this r ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ h : r = 0 ⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ ** simp [h] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ h : ¬r = 0 rinv : r * r⁻¹ = 1 ⊢ r⁻¹ * r = 1 ** rw [mul_comm] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ h : ¬r = 0 rinv : r * r⁻¹ = 1 ⊢ r * r⁻¹ = 1 ** exact rinv ** Qed
| |
MeasureTheory.lintegral_mul_const ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r ** simp_rw [mul_comm, lintegral_const_mul r hf] ** Qed
| |
MeasureTheory.lintegral_mul_const'' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hf : AEMeasurable f ⊢ ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r ** simp_rw [mul_comm, lintegral_const_mul'' r hf] ** Qed
| |
MeasureTheory.lintegral_mul_const_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ ⊢ (∫⁻ (a : α), f a ∂μ) * r ≤ ∫⁻ (a : α), f a * r ∂μ ** simp_rw [mul_comm, lintegral_const_mul_le r f] ** Qed
| |
MeasureTheory.lintegral_mul_const' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α r : ℝ≥0∞ f : α → ℝ≥0∞ hr : r ≠ ⊤ ⊢ ∫⁻ (a : α), f a * r ∂μ = (∫⁻ (a : α), f a ∂μ) * r ** simp_rw [mul_comm, lintegral_const_mul' r f hr] ** Qed
| |
MeasureTheory.lintegral_lintegral_mul ** α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν✝ : Measure α β : Type u_5 inst✝ : MeasurableSpace β ν : Measure β f : α → ℝ≥0∞ g : β → ℝ≥0∞ hf : AEMeasurable f hg : AEMeasurable g ⊢ ∫⁻ (x : α), ∫⁻ (y : β), f x * g y ∂ν ∂μ = (∫⁻ (x : α), f x ∂μ) * ∫⁻ (y : β), g y ∂ν ** simp [lintegral_const_mul'' _ hg, lintegral_mul_const'' _ hf] ** Qed
| |
MeasureTheory.lintegral_rw₁ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f f' : α → β h✝ : f =ᵐ[μ] f' g : β → ℝ≥0∞ a : α h : f a = f' a ⊢ (fun a => g (f a)) a = (fun a => g (f' a)) a ** dsimp only ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f f' : α → β h✝ : f =ᵐ[μ] f' g : β → ℝ≥0∞ a : α h : f a = f' a ⊢ g (f a) = g (f' a) ** rw [h] ** Qed
| |
MeasureTheory.lintegral_rw₂ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f₁ f₁' : α → β f₂ f₂' : α → γ h₁✝ : f₁ =ᵐ[μ] f₁' h₂✝ : f₂ =ᵐ[μ] f₂' g : β → γ → ℝ≥0∞ x✝ : α h₂ : f₂ x✝ = f₂' x✝ h₁ : f₁ x✝ = f₁' x✝ ⊢ (fun a => g (f₁ a) (f₂ a)) x✝ = (fun a => g (f₁' a) (f₂' a)) x✝ ** dsimp only ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f₁ f₁' : α → β f₂ f₂' : α → γ h₁✝ : f₁ =ᵐ[μ] f₁' h₂✝ : f₂ =ᵐ[μ] f₂' g : β → γ → ℝ≥0∞ x✝ : α h₂ : f₂ x✝ = f₂' x✝ h₁ : f₁ x✝ = f₁' x✝ ⊢ g (f₁ x✝) (f₂ x✝) = g (f₁' x✝) (f₂' x✝) ** rw [h₁, h₂] ** Qed
| |
MeasureTheory.lintegral_indicator ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α), indicator s f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ ** simp only [lintegral, ← restrict_lintegral_eq_lintegral_restrict _ hs, iSup_subtype'] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s ⊢ ⨆ x, SimpleFunc.lintegral (↑x) μ = ⨆ x, SimpleFunc.lintegral (restrict (↑x) s) μ ** apply le_antisymm <;> refine' iSup_mono' (Subtype.forall.2 fun φ hφ => _) ** case a α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s φ : α →ₛ ℝ≥0∞ hφ : ↑φ ≤ fun a => indicator s f a ⊢ ∃ i', SimpleFunc.lintegral (↑{ val := φ, property := hφ }) μ ≤ SimpleFunc.lintegral (restrict (↑i') s) μ ** refine' ⟨⟨φ, le_trans hφ (indicator_le_self _ _)⟩, _⟩ ** case a α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s φ : α →ₛ ℝ≥0∞ hφ : ↑φ ≤ fun a => indicator s f a ⊢ SimpleFunc.lintegral (↑{ val := φ, property := hφ }) μ ≤ SimpleFunc.lintegral (restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) μ ** refine' SimpleFunc.lintegral_mono (fun x => _) le_rfl ** case a α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s φ : α →ₛ ℝ≥0∞ hφ : ↑φ ≤ fun a => indicator s f a x : α ⊢ ↑↑{ val := φ, property := hφ } x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x ** by_cases hx : x ∈ s ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s φ : α →ₛ ℝ≥0∞ hφ : ↑φ ≤ fun a => indicator s f a x : α hx : x ∈ s ⊢ ↑↑{ val := φ, property := hφ } x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x ** simp [hx, hs, le_refl] ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s φ : α →ₛ ℝ≥0∞ hφ : ↑φ ≤ fun a => indicator s f a x : α hx : ¬x ∈ s ⊢ ↑↑{ val := φ, property := hφ } x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x ** apply le_trans (hφ x) ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s φ : α →ₛ ℝ≥0∞ hφ : ↑φ ≤ fun a => indicator s f a x : α hx : ¬x ∈ s ⊢ (fun a => indicator s f a) x ≤ ↑(restrict (↑{ val := φ, property := (_ : ↑φ ≤ fun a => f a) }) s) x ** simp [hx, hs, le_refl] ** case a α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s φ : α →ₛ ℝ≥0∞ hφ : ↑φ ≤ fun a => f a ⊢ ∃ i', SimpleFunc.lintegral (restrict (↑{ val := φ, property := hφ }) s) μ ≤ SimpleFunc.lintegral (↑i') μ ** refine' ⟨⟨φ.restrict s, fun x => _⟩, le_rfl⟩ ** case a α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : MeasurableSet s φ : α →ₛ ℝ≥0∞ hφ : ↑φ ≤ fun a => f a x : α ⊢ ↑(restrict φ s) x ≤ (fun a => indicator s f a) x ** simp [hφ x, hs, indicator_le_indicator] ** Qed
| |
MeasureTheory.lintegral_indicator₀ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hs : NullMeasurableSet s ⊢ ∫⁻ (a : α), indicator s f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ ** rw [← lintegral_congr_ae (indicator_ae_eq_of_ae_eq_set hs.toMeasurable_ae_eq),
lintegral_indicator _ (measurableSet_toMeasurable _ _),
Measure.restrict_congr_set hs.toMeasurable_ae_eq] ** Qed
| |
MeasureTheory.lintegral_indicator_const₀ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α hs : NullMeasurableSet s c : ℝ≥0∞ ⊢ ∫⁻ (a : α), indicator s (fun x => c) a ∂μ = c * ↑↑μ s ** rw [lintegral_indicator₀ _ hs, set_lintegral_const] ** Qed
| |
MeasureTheory.set_lintegral_eq_const ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f r : ℝ≥0∞ ⊢ ∫⁻ (x : α) in {x | f x = r}, f x ∂μ = r * ↑↑μ {x | f x = r} ** have : ∀ᵐ x ∂μ, x ∈ { x | f x = r } → f x = r := ae_of_all μ fun _ hx => hx ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f r : ℝ≥0∞ this : ∀ᵐ (x : α) ∂μ, x ∈ {x | f x = r} → f x = r ⊢ ∫⁻ (x : α) in {x | f x = r}, f x ∂μ = r * ↑↑μ {x | f x = r} ** rw [set_lintegral_congr_fun _ this] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f r : ℝ≥0∞ this : ∀ᵐ (x : α) ∂μ, x ∈ {x | f x = r} → f x = r ⊢ ∫⁻ (x : α) in {x | f x = r}, r ∂μ = r * ↑↑μ {x | f x = r} α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f r : ℝ≥0∞ this : ∀ᵐ (x : α) ∂μ, x ∈ {x | f x = r} → f x = r ⊢ MeasurableSet {x | f x = r} ** rw [lintegral_const, Measure.restrict_apply MeasurableSet.univ, Set.univ_inter] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f r : ℝ≥0∞ this : ∀ᵐ (x : α) ∂μ, x ∈ {x | f x = r} → f x = r ⊢ MeasurableSet {x | f x = r} ** exact hf (measurableSet_singleton r) ** Qed
| |
MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ ⊢ ∫⁻ (a : α), f a ∂μ + ε * ↑↑μ {x | f x + ε ≤ g x} ≤ ∫⁻ (a : α), g a ∂μ ** rcases exists_measurable_le_lintegral_eq μ f with ⟨φ, hφm, hφ_le, hφ_eq⟩ ** case intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ ⊢ ∫⁻ (a : α), f a ∂μ + ε * ↑↑μ {x | f x + ε ≤ g x} ≤ ∫⁻ (a : α), g a ∂μ ** calc
∫⁻ x, f x ∂μ + ε * μ { x | f x + ε ≤ g x } = ∫⁻ x, φ x ∂μ + ε * μ { x | f x + ε ≤ g x } :=
by rw [hφ_eq]
_ ≤ ∫⁻ x, φ x ∂μ + ε * μ { x | φ x + ε ≤ g x } := by
gcongr
exact measure_mono fun x => (add_le_add_right (hφ_le _) _).trans
_ = ∫⁻ x, φ x + indicator { x | φ x + ε ≤ g x } (fun _ => ε) x ∂μ := by
rw [lintegral_add_left hφm, lintegral_indicator₀, set_lintegral_const]
exact measurableSet_le (hφm.nullMeasurable.measurable'.add_const _) hg.nullMeasurable
_ ≤ ∫⁻ x, g x ∂μ := lintegral_mono_ae (hle.mono fun x hx₁ => ?_) ** case intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ x : α hx₁ : f x ≤ g x ⊢ φ x + indicator {x | φ x + ε ≤ g x} (fun x => ε) x ≤ g x ** simp only [indicator_apply] ** case intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ x : α hx₁ : f x ≤ g x ⊢ (φ x + if x ∈ {x | φ x + ε ≤ g x} then ε else 0) ≤ g x ** split_ifs with hx₂ ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ x : α hx₁ : f x ≤ g x hx₂ : x ∈ {x | φ x + ε ≤ g x} ⊢ φ x + ε ≤ g x case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ x : α hx₁ : f x ≤ g x hx₂ : ¬x ∈ {x | φ x + ε ≤ g x} ⊢ φ x + 0 ≤ g x ** exacts [hx₂, (add_zero _).trans_le <| (hφ_le x).trans hx₁] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ ⊢ ∫⁻ (x : α), f x ∂μ + ε * ↑↑μ {x | f x + ε ≤ g x} = ∫⁻ (x : α), φ x ∂μ + ε * ↑↑μ {x | f x + ε ≤ g x} ** rw [hφ_eq] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ ⊢ ∫⁻ (x : α), φ x ∂μ + ε * ↑↑μ {x | f x + ε ≤ g x} ≤ ∫⁻ (x : α), φ x ∂μ + ε * ↑↑μ {x | φ x + ε ≤ g x} ** gcongr ** case bc.bc α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ ⊢ ↑↑μ {x | f x + ε ≤ g x} ≤ ↑↑μ {x | φ x + ε ≤ g x} ** exact measure_mono fun x => (add_le_add_right (hφ_le _) _).trans ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ ⊢ ∫⁻ (x : α), φ x ∂μ + ε * ↑↑μ {x | φ x + ε ≤ g x} = ∫⁻ (x : α), φ x + indicator {x | φ x + ε ≤ g x} (fun x => ε) x ∂μ ** rw [lintegral_add_left hφm, lintegral_indicator₀, set_lintegral_const] ** case hs α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hle : f ≤ᵐ[μ] g hg : AEMeasurable g ε : ℝ≥0∞ φ : α → ℝ≥0∞ hφm : Measurable φ hφ_le : φ ≤ f hφ_eq : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), φ a ∂μ ⊢ NullMeasurableSet {x | φ x + ε ≤ g x} ** exact measurableSet_le (hφm.nullMeasurable.measurable'.add_const _) hg.nullMeasurable ** Qed
| |
MeasureTheory.mul_meas_ge_le_lintegral₀ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f ε : ℝ≥0∞ ⊢ ε * ↑↑μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ ** simpa only [lintegral_zero, zero_add] using
lintegral_add_mul_meas_add_le_le_lintegral (ae_of_all _ fun x => zero_le (f x)) hf ε ** Qed
| |
MeasureTheory.lintegral_eq_top_of_measure_eq_top_ne_zero ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f hμf : ↑↑μ {x | f x = ⊤} ≠ 0 ⊢ ⊤ = ⊤ * ↑↑μ {x | ⊤ ≤ f x} ** simp [mul_eq_top, hμf] ** Qed
| |
MeasureTheory.setLintegral_eq_top_of_measure_eq_top_ne_zero ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hf : AEMeasurable f hμf : ↑↑μ {x | x ∈ s ∧ f x = ⊤} ≠ 0 ⊢ ↑↑μ {x | x ∈ s ∧ f x = ⊤} ≤ ↑↑(Measure.restrict μ s) {x | f x = ⊤} ** rw [←setOf_inter_eq_sep] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s : Set α hf : AEMeasurable f hμf : ↑↑μ {x | x ∈ s ∧ f x = ⊤} ≠ 0 ⊢ ↑↑μ ({a | f a = ⊤} ∩ s) ≤ ↑↑(Measure.restrict μ s) {x | f x = ⊤} ** exact Measure.le_restrict_apply _ _ ** Qed
| |
MeasureTheory.meas_ge_le_lintegral_div ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f ε : ℝ≥0∞ hε : ε ≠ 0 hε' : ε ≠ ⊤ ⊢ ↑↑μ {x | ε ≤ f x} * ε ≤ ∫⁻ (a : α), f a ∂μ ** rw [mul_comm] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f ε : ℝ≥0∞ hε : ε ≠ 0 hε' : ε ≠ ⊤ ⊢ ε * ↑↑μ {x | ε ≤ f x} ≤ ∫⁻ (a : α), f a ∂μ ** exact mul_meas_ge_le_lintegral₀ hf ε ** Qed
| |
MeasureTheory.lintegral_eq_zero_iff' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f ⊢ ∫⁻ (x : α), 0 ∂μ ≠ ⊤ ** simp [lintegral_zero, zero_ne_top] ** Qed
| |
MeasureTheory.lintegral_pos_iff_support ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f ⊢ 0 < ∫⁻ (a : α), f a ∂μ ↔ 0 < ↑↑μ (support f) ** simp [pos_iff_ne_zero, hf, Filter.EventuallyEq, ae_iff, Function.support] ** Qed
| |
MeasureTheory.lintegral_iSup_ae ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a s : Set α hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ∧ MeasurableSet s ∧ ↑↑μ s = 0 g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a a : α ha : ∀ (n : ℕ), g n a = f n a ⊢ (fun a => ⨆ n, f n a) a = (fun a => ⨆ n, g n a) a ** simp only [ha] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a s : Set α hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ∧ MeasurableSet s ∧ ↑↑μ s = 0 g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a n : ℕ a : α h : a ∈ s ⊢ g n a ≤ g (n + 1) a ** simp [if_pos h] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a s : Set α hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ∧ MeasurableSet s ∧ ↑↑μ s = 0 g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a n : ℕ a : α h : ¬a ∈ s ⊢ g n a ≤ g (n + 1) a ** simp only [if_neg h] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a s : Set α hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ∧ MeasurableSet s ∧ ↑↑μ s = 0 g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a n : ℕ a : α h : ¬a ∈ s ⊢ f n a ≤ f (n + 1) a ** have := hs.1 ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a s : Set α hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ∧ MeasurableSet s ∧ ↑↑μ s = 0 g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a n : ℕ a : α h : ¬a ∈ s this : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ⊢ f n a ≤ f (n + 1) a ** rw [subset_def] at this ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a s : Set α hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ∧ MeasurableSet s ∧ ↑↑μ s = 0 g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a n : ℕ a : α h : ¬a ∈ s this : ∀ (x : α), x ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} → x ∈ s ⊢ f n a ≤ f (n + 1) a ** have := mt (this a) h ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a s : Set α hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ∧ MeasurableSet s ∧ ↑↑μ s = 0 g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a n : ℕ a : α h : ¬a ∈ s this✝ : ∀ (x : α), x ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} → x ∈ s this : ¬a ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊢ f n a ≤ f (n + 1) a ** simp only [Classical.not_not, mem_setOf_eq] at this ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a s : Set α hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ∧ MeasurableSet s ∧ ↑↑μ s = 0 g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a n : ℕ a : α h : ¬a ∈ s this✝ : ∀ (x : α), x ∈ {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} → x ∈ s this : ∀ (i : ℕ), f i a ≤ f (Nat.succ i) a ⊢ f n a ≤ f (n + 1) a ** exact this n ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f (Nat.succ n) a s : Set α hs : {a | ¬∀ (i : ℕ), f i a ≤ f (Nat.succ i) a} ⊆ s ∧ MeasurableSet s ∧ ↑↑μ s = 0 g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a ⊢ ⨆ n, ∫⁻ (a : α), g n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ ** simp only [lintegral_congr_ae (g_eq_f.mono fun _a ha => ha _)] ** Qed
| |
MeasureTheory.lintegral_sub' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hg : AEMeasurable g hg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤ h_le : g ≤ᵐ[μ] f ⊢ ∫⁻ (a : α), f a - g a ∂μ = ∫⁻ (a : α), f a ∂μ - ∫⁻ (a : α), g a ∂μ ** refine' ENNReal.eq_sub_of_add_eq hg_fin _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hg : AEMeasurable g hg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤ h_le : g ≤ᵐ[μ] f ⊢ ∫⁻ (a : α), f a - g a ∂μ + ∫⁻ (a : α), g a ∂μ = ∫⁻ (a : α), f a ∂μ ** rw [← lintegral_add_right' _ hg] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hg : AEMeasurable g hg_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤ h_le : g ≤ᵐ[μ] f ⊢ ∫⁻ (a : α), f a - g a + g a ∂μ = ∫⁻ (a : α), f a ∂μ ** exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx) ** Qed
| |
MeasureTheory.lintegral_sub_le' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f ⊢ ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α), f x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ ** rw [tsub_le_iff_right] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f ⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ ** by_cases hfi : ∫⁻ x, f x ∂μ = ∞ ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hfi : ∫⁻ (x : α), f x ∂μ = ⊤ ⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ ** rw [hfi, add_top] ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hfi : ∫⁻ (x : α), f x ∂μ = ⊤ ⊢ ∫⁻ (x : α), g x ∂μ ≤ ⊤ ** exact le_top ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hfi : ¬∫⁻ (x : α), f x ∂μ = ⊤ ⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ ** rw [← lintegral_add_right' _ hf] ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : AEMeasurable f hfi : ¬∫⁻ (x : α), f x ∂μ = ⊤ ⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (a : α), g a - f a + f a ∂μ ** exact lintegral_mono fun x => le_tsub_add ** Qed
| |
MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hg : AEMeasurable g hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h_le : f ≤ᵐ[μ] g h : ∃ᵐ (x : α) ∂μ, f x ≠ g x ⊢ ∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ ** contrapose! h ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hg : AEMeasurable g hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h_le : f ≤ᵐ[μ] g h : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ ⊢ ¬∃ᵐ (x : α) ∂μ, f x ≠ g x ** simp only [not_frequently, Ne.def, Classical.not_not] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hg : AEMeasurable g hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h_le : f ≤ᵐ[μ] g h : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ ⊢ ∀ᵐ (x : α) ∂μ, f x = g x ** exact ae_eq_of_ae_le_of_lintegral_le h_le hfi hg h ** Qed
| |
MeasureTheory.tendsto_lintegral_of_dominated_convergence' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) ⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** have : ∀ n, ∫⁻ a, F n a ∂μ = ∫⁻ a, (hF_meas n).mk (F n) a ∂μ := fun n =>
lintegral_congr_ae (hF_meas n).ae_eq_mk ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ ⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** simp_rw [this] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ ⊢ Tendsto (fun n => ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** apply
tendsto_lintegral_of_dominated_convergence bound (fun n => (hF_meas n).measurable_mk) _ h_fin ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ ⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a)) ** have : ∀ n, ∀ᵐ a ∂μ, (hF_meas n).mk (F n) a = F n a := fun n => (hF_meas n).ae_eq_mk.symm ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this✝ : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ this : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a ⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a)) ** have : ∀ᵐ a ∂μ, ∀ n, (hF_meas n).mk (F n) a = F n a := ae_all_iff.mpr this ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this✝¹ : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ this✝ : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a this : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a ⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a)) ** filter_upwards [this, h_lim] with a H H' ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this✝¹ : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ this✝ : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a this : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a a : α H : ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a H' : Tendsto (fun n => F n a) atTop (𝓝 (f a)) ⊢ Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (𝓝 (f a)) ** simp_rw [H] ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this✝¹ : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ this✝ : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a this : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a a : α H : ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a H' : Tendsto (fun n => F n a) atTop (𝓝 (f a)) ⊢ Tendsto (fun n => F n a) atTop (𝓝 (f a)) ** exact H' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ ⊢ ∀ (n : ℕ), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) ≤ᵐ[μ] bound ** intro n ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ n : ℕ ⊢ AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) ≤ᵐ[μ] bound ** filter_upwards [h_bound n, (hF_meas n).ae_eq_mk] with a H H' ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α F : ℕ → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ (n : ℕ), AEMeasurable (F n) h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a)) this : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ∂μ n : ℕ a : α H : F n a ≤ bound a H' : F n a = AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ⊢ AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a ≤ bound a ** rwa [H'] at H ** Qed
| |
MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) ⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) l (𝓝 (∫⁻ (a : α), f a ∂μ)) ** rw [tendsto_iff_seq_tendsto] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) ⊢ ∀ (x : ℕ → ι), Tendsto x atTop l → Tendsto ((fun n => ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** intro x xl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l ⊢ Tendsto ((fun n => ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** have hxl := by
rw [tendsto_atTop'] at xl
exact xl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s ⊢ Tendsto ((fun n => ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** have h := inter_mem hF_meas h_bound ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s h : {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ∈ l ⊢ Tendsto ((fun n => ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** replace h := hxl _ h ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s h : ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ⊢ Tendsto ((fun n => ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** rcases h with ⟨k, h⟩ ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ⊢ Tendsto ((fun n => ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** rw [← tendsto_add_atTop_iff_nat k] ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ⊢ Tendsto (fun n => ((fun n => ∫⁻ (a : α), F n a ∂μ) ∘ x) (n + k)) atTop (𝓝 (∫⁻ (a : α), f a ∂μ)) ** refine' tendsto_lintegral_of_dominated_convergence _ _ _ _ _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l ⊢ ?m.1148951 ** rw [tendsto_atTop'] at xl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl✝ : Tendsto x atTop l xl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s ⊢ ?m.1148951 ** exact xl ** case intro.refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ⊢ α → ℝ≥0∞ ** exact bound ** case intro.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ⊢ ∀ (n : ℕ), Measurable fun a => F (x (n + k)) a ** intro ** case intro.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} n✝ : ℕ ⊢ Measurable fun a => F (x (n✝ + k)) a ** refine' (h _ _).1 ** case intro.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} n✝ : ℕ ⊢ n✝ + k ≥ k ** exact Nat.le_add_left _ _ ** case intro.refine'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ⊢ ∀ (n : ℕ), (fun a => F (x (n + k)) a) ≤ᵐ[μ] bound ** intro ** case intro.refine'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} n✝ : ℕ ⊢ (fun a => F (x (n✝ + k)) a) ≤ᵐ[μ] bound ** refine' (h _ _).2 ** case intro.refine'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} n✝ : ℕ ⊢ n✝ + k ≥ k ** exact Nat.le_add_left _ _ ** case intro.refine'_4 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ⊢ ∫⁻ (a : α), bound a ∂μ ≠ ⊤ ** assumption ** case intro.refine'_5 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} ⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F (x (n + k)) a) atTop (𝓝 (f a)) ** refine' h_lim.mono fun a h_lim => _ ** case intro.refine'_5 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim✝ : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} a : α h_lim : Tendsto (fun n => F n a) l (𝓝 (f a)) ⊢ Tendsto (fun n => F (x (n + k)) a) atTop (𝓝 (f a)) ** apply @Tendsto.comp _ _ _ (fun n => x (n + k)) fun n => F n a ** case intro.refine'_5.hf α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim✝ : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} a : α h_lim : Tendsto (fun n => F n a) l (𝓝 (f a)) ⊢ Tendsto (fun n => x (n + k)) atTop l ** rw [tendsto_add_atTop_iff_nat] ** case intro.refine'_5.hf α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim✝ : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} a : α h_lim : Tendsto (fun n => F n a) l (𝓝 (f a)) ⊢ Tendsto x atTop l ** assumption ** case intro.refine'_5.hg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α ι : Type u_5 l : Filter ι inst✝ : IsCountablyGenerated l F : ι → α → ℝ≥0∞ f bound : α → ℝ≥0∞ hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n) h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤ h_lim✝ : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a)) x : ℕ → ι xl : Tendsto x atTop l hxl : ∀ (s : Set ι), s ∈ l → ∃ a, ∀ (b : ℕ), b ≥ a → x b ∈ s k : ℕ h : ∀ (b : ℕ), b ≥ k → x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x} a : α h_lim : Tendsto (fun n => F n a) l (𝓝 (f a)) ⊢ Tendsto (fun n => F n a) ?intro.refine'_5.y (𝓝 (f a)) ** assumption ** Qed
| |
MeasureTheory.lintegral_iSup_directed_of_measurable ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f ⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ ** cases nonempty_encodable β ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β ⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ ** cases isEmpty_or_nonempty β ** case intro.inr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : Nonempty β ⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ ** inhabit β ** case intro.inr α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : Nonempty β inhabited_h : Inhabited β ⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ ** have : ∀ a, ⨆ b, f b a = ⨆ n, f (h_directed.sequence f n) a := by
intro a
refine' le_antisymm (iSup_le fun b => _) (iSup_le fun n => le_iSup (fun n => f n a) _)
exact le_iSup_of_le (encode b + 1) (h_directed.le_sequence b a) ** case intro.inl α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : IsEmpty β ⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ ** simp [iSup_of_empty] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : Nonempty β inhabited_h : Inhabited β ⊢ ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a ** intro a ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : Nonempty β inhabited_h : Inhabited β a : α ⊢ ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a ** refine' le_antisymm (iSup_le fun b => _) (iSup_le fun n => le_iSup (fun n => f n a) _) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : Nonempty β inhabited_h : Inhabited β a : α b : β ⊢ f b a ≤ ⨆ n, f (Directed.sequence f h_directed n) a ** exact le_iSup_of_le (encode b + 1) (h_directed.le_sequence b a) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : Nonempty β inhabited_h : Inhabited β this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a ⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ∫⁻ (a : α), ⨆ n, f (Directed.sequence f h_directed n) a ∂μ ** simp only [this] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : Nonempty β inhabited_h : Inhabited β this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a ⊢ ⨆ n, ∫⁻ (a : α), f (Directed.sequence f h_directed n) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ ** refine' le_antisymm (iSup_le fun n => _) (iSup_le fun b => _) ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : Nonempty β inhabited_h : Inhabited β this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a n : ℕ ⊢ ∫⁻ (a : α), f (Directed.sequence f h_directed n) a ∂μ ≤ ⨆ b, ∫⁻ (a : α), f b a ∂μ ** exact le_iSup (fun b => ∫⁻ a, f b a ∂μ) _ ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β f : β → α → ℝ≥0∞ hf : ∀ (b : β), Measurable (f b) h_directed : Directed (fun x x_1 => x ≤ x_1) f val✝ : Encodable β h✝ : Nonempty β inhabited_h : Inhabited β this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a b : β ⊢ ∫⁻ (a : α), f b a ∂μ ≤ ⨆ n, ∫⁻ (a : α), f (Directed.sequence f h_directed n) a ∂μ ** exact le_iSup_of_le (encode b + 1) (lintegral_mono <| h_directed.le_sequence b) ** Qed
| |
MeasureTheory.lintegral_iUnion₀ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β s : β → Set α hm : ∀ (i : β), NullMeasurableSet (s i) hd : Pairwise (AEDisjoint μ on s) f : α → ℝ≥0∞ ⊢ ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ = ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ ** simp only [Measure.restrict_iUnion_ae hd hm, lintegral_sum_measure] ** Qed
| |
MeasureTheory.lintegral_biUnion₀ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α t : Set β s : β → Set α ht : Set.Countable t hm : ∀ (i : β), i ∈ t → NullMeasurableSet (s i) hd : Set.Pairwise t (AEDisjoint μ on s) f : α → ℝ≥0∞ ⊢ ∫⁻ (a : α) in ⋃ i ∈ t, s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ ** haveI := ht.toEncodable ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α t : Set β s : β → Set α ht : Set.Countable t hm : ∀ (i : β), i ∈ t → NullMeasurableSet (s i) hd : Set.Pairwise t (AEDisjoint μ on s) f : α → ℝ≥0∞ this : Encodable ↑t ⊢ ∫⁻ (a : α) in ⋃ i ∈ t, s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ ** rw [biUnion_eq_iUnion, lintegral_iUnion₀ (SetCoe.forall'.1 hm) (hd.subtype _ _)] ** Qed
| |
MeasureTheory.lintegral_biUnion_finset₀ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Finset β t : β → Set α hd : Set.Pairwise (↑s) (AEDisjoint μ on t) hm : ∀ (b : β), b ∈ s → NullMeasurableSet (t b) f : α → ℝ≥0∞ ⊢ ∫⁻ (a : α) in ⋃ b ∈ s, t b, f a ∂μ = ∑ b in s, ∫⁻ (a : α) in t b, f a ∂μ ** simp only [← Finset.mem_coe, lintegral_biUnion₀ s.countable_toSet hm hd, ← Finset.tsum_subtype'] ** Qed
| |
MeasureTheory.lintegral_iUnion_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β s : β → Set α f : α → ℝ≥0∞ ⊢ ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ ≤ ∑' (i : β), ∫⁻ (a : α) in s i, f a ∂μ ** rw [← lintegral_sum_measure] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : Countable β s : β → Set α f : α → ℝ≥0∞ ⊢ ∫⁻ (a : α) in ⋃ i, s i, f a ∂μ ≤ ∫⁻ (a : α), f a ∂sum fun i => Measure.restrict μ (s i) ** exact lintegral_mono' restrict_iUnion_le le_rfl ** Qed
| |
MeasureTheory.lintegral_union ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ A B : Set α hB : MeasurableSet B hAB : Disjoint A B ⊢ ∫⁻ (a : α) in A ∪ B, f a ∂μ = ∫⁻ (a : α) in A, f a ∂μ + ∫⁻ (a : α) in B, f a ∂μ ** rw [restrict_union hAB hB, lintegral_add_measure] ** Qed
| |
MeasureTheory.lintegral_union_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s t : Set α ⊢ ∫⁻ (a : α) in s ∪ t, f a ∂μ ≤ ∫⁻ (a : α) in s, f a ∂μ + ∫⁻ (a : α) in t, f a ∂μ ** rw [← lintegral_add_measure] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ s t : Set α ⊢ ∫⁻ (a : α) in s ∪ t, f a ∂μ ≤ ∫⁻ (a : α), f a ∂(Measure.restrict μ s + Measure.restrict μ t) ** exact lintegral_mono' (restrict_union_le _ _) le_rfl ** Qed
| |
MeasureTheory.lintegral_add_compl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ A : Set α hA : MeasurableSet A ⊢ ∫⁻ (x : α) in A, f x ∂μ + ∫⁻ (x : α) in Aᶜ, f x ∂μ = ∫⁻ (x : α), f x ∂μ ** rw [← lintegral_add_measure, Measure.restrict_add_restrict_compl hA] ** Qed
| |
MeasureTheory.lintegral_max ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : Measurable f hg : Measurable g ⊢ ∫⁻ (x : α), max (f x) (g x) ∂μ = ∫⁻ (x : α) in {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in {x | g x < f x}, f x ∂μ ** have hm : MeasurableSet { x | f x ≤ g x } := measurableSet_le hf hg ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : Measurable f hg : Measurable g hm : MeasurableSet {x | f x ≤ g x} ⊢ ∫⁻ (x : α), max (f x) (g x) ∂μ = ∫⁻ (x : α) in {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in {x | g x < f x}, f x ∂μ ** rw [← lintegral_add_compl (fun x => max (f x) (g x)) hm] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : Measurable f hg : Measurable g hm : MeasurableSet {x | f x ≤ g x} ⊢ ∫⁻ (x : α) in {x | f x ≤ g x}, max (f x) (g x) ∂μ + ∫⁻ (x : α) in {x | f x ≤ g x}ᶜ, max (f x) (g x) ∂μ = ∫⁻ (x : α) in {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in {x | g x < f x}, f x ∂μ ** simp only [← compl_setOf, ← not_le] ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : Measurable f hg : Measurable g hm : MeasurableSet {x | f x ≤ g x} ⊢ ∀ᵐ (x : α) ∂μ, x ∈ {x | f x ≤ g x} → max (f x) (g x) = g x case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f g : α → ℝ≥0∞ hf : Measurable f hg : Measurable g hm : MeasurableSet {x | f x ≤ g x} ⊢ ∀ᵐ (x : α) ∂μ, x ∈ {x | f x ≤ g x}ᶜ → max (f x) (g x) = f x ** exacts [ae_of_all _ fun x => max_eq_right (a := f x) (b := g x),
ae_of_all _ fun x (hx : ¬ f x ≤ g x) => max_eq_left (not_le.1 hx).le] ** Qed
| |
MeasureTheory.lintegral_map' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α mβ : MeasurableSpace β f : β → ℝ≥0∞ g : α → β hf : AEMeasurable f hg : AEMeasurable g ⊢ ∫⁻ (a : β), AEMeasurable.mk f hf a ∂Measure.map g μ = ∫⁻ (a : β), AEMeasurable.mk f hf a ∂Measure.map (AEMeasurable.mk g hg) μ ** congr 1 ** case e_μ α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α mβ : MeasurableSpace β f : β → ℝ≥0∞ g : α → β hf : AEMeasurable f hg : AEMeasurable g ⊢ Measure.map g μ = Measure.map (AEMeasurable.mk g hg) μ ** exact Measure.map_congr hg.ae_eq_mk ** Qed
| |
MeasureTheory.lintegral_map_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α mβ : MeasurableSpace β f : β → ℝ≥0∞ g : α → β hg : Measurable g ⊢ ∫⁻ (a : β), f a ∂Measure.map g μ ≤ ∫⁻ (a : α), f (g a) ∂μ ** rw [← iSup_lintegral_measurable_le_eq_lintegral, ← iSup_lintegral_measurable_le_eq_lintegral] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α mβ : MeasurableSpace β f : β → ℝ≥0∞ g : α → β hg : Measurable g ⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => f a), ∫⁻ (a : β), g_1 a ∂Measure.map g μ ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => f (g a)), ∫⁻ (a : α), g_1 a ∂μ ** refine' iSup₂_le fun i hi => iSup_le fun h'i => _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α mβ : MeasurableSpace β f : β → ℝ≥0∞ g : α → β hg : Measurable g i : β → ℝ≥0∞ hi : Measurable i h'i : i ≤ fun a => f a ⊢ ∫⁻ (a : β), i a ∂Measure.map g μ ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ fun a => f (g a)), ∫⁻ (a : α), g_1 a ∂μ ** refine' le_iSup₂_of_le (i ∘ g) (hi.comp hg) _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α mβ : MeasurableSpace β f : β → ℝ≥0∞ g : α → β hg : Measurable g i : β → ℝ≥0∞ hi : Measurable i h'i : i ≤ fun a => f a ⊢ ∫⁻ (a : β), i a ∂Measure.map g μ ≤ ⨆ (_ : i ∘ g ≤ fun a => f (g a)), ∫⁻ (a : α), (i ∘ g) a ∂μ ** exact le_iSup_of_le (fun x => h'i (g x)) (le_of_eq (lintegral_map hi hg)) ** Qed
| |
MeasureTheory.set_lintegral_map ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β f : β → ℝ≥0∞ g : α → β s : Set β hs : MeasurableSet s hf : Measurable f hg : Measurable g ⊢ ∫⁻ (y : β) in s, f y ∂Measure.map g μ = ∫⁻ (x : α) in g ⁻¹' s, f (g x) ∂μ ** rw [restrict_map hg hs, lintegral_map hf hg] ** Qed
| |
MeasureTheory.lintegral_indicator_const_comp ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α mβ : MeasurableSpace β f : α → β s : Set β hf : Measurable f hs : MeasurableSet s c : ℝ≥0∞ ⊢ ∫⁻ (a : α), indicator s (fun x => c) (f a) ∂μ = c * ↑↑μ (f ⁻¹' s) ** erw [lintegral_comp (measurable_const.indicator hs) hf, lintegral_indicator_const hs,
Measure.map_apply hf hs] ** Qed
| |
MeasurableEmbedding.lintegral_map ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β g : α → β hg : MeasurableEmbedding g f : β → ℝ≥0∞ ⊢ ∫⁻ (a : β), f a ∂Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ ** rw [lintegral, lintegral] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β g : α → β hg : MeasurableEmbedding g f : β → ℝ≥0∞ ⊢ ⨆ g_1, ⨆ (_ : ↑g_1 ≤ fun a => f a), SimpleFunc.lintegral g_1 (Measure.map g μ) = ⨆ g_1, ⨆ (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ ** refine' le_antisymm (iSup₂_le fun f₀ hf₀ => _) (iSup₂_le fun f₀ hf₀ => _) ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β g : α → β hg : MeasurableEmbedding g f : β → ℝ≥0∞ f₀ : β →ₛ ℝ≥0∞ hf₀ : ↑f₀ ≤ fun a => f a ⊢ SimpleFunc.lintegral f₀ (Measure.map g μ) ≤ ⨆ g_1, ⨆ (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ ** rw [SimpleFunc.lintegral_map _ hg.measurable] ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β g : α → β hg : MeasurableEmbedding g f : β → ℝ≥0∞ f₀ : β →ₛ ℝ≥0∞ hf₀ : ↑f₀ ≤ fun a => f a ⊢ SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ ≤ ⨆ g_1, ⨆ (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ ** have : (f₀.comp g hg.measurable : α → ℝ≥0∞) ≤ f ∘ g := fun x => hf₀ (g x) ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β g : α → β hg : MeasurableEmbedding g f : β → ℝ≥0∞ f₀ : β →ₛ ℝ≥0∞ hf₀ : ↑f₀ ≤ fun a => f a this : ↑(comp f₀ g (_ : Measurable g)) ≤ f ∘ g ⊢ SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ ≤ ⨆ g_1, ⨆ (_ : ↑g_1 ≤ fun a => f (g a)), SimpleFunc.lintegral g_1 μ ** exact le_iSup_of_le (comp f₀ g hg.measurable) (by exact le_iSup (α := ℝ≥0∞) _ this) ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β g : α → β hg : MeasurableEmbedding g f : β → ℝ≥0∞ f₀ : β →ₛ ℝ≥0∞ hf₀ : ↑f₀ ≤ fun a => f a this : ↑(comp f₀ g (_ : Measurable g)) ≤ f ∘ g ⊢ SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ ≤ ⨆ (_ : ↑(comp f₀ g (_ : Measurable g)) ≤ fun a => f (g a)), SimpleFunc.lintegral (comp f₀ g (_ : Measurable g)) μ ** exact le_iSup (α := ℝ≥0∞) _ this ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β g : α → β hg : MeasurableEmbedding g f : β → ℝ≥0∞ f₀ : α →ₛ ℝ≥0∞ hf₀ : ↑f₀ ≤ fun a => f (g a) ⊢ SimpleFunc.lintegral f₀ μ ≤ ⨆ g_1, ⨆ (_ : ↑g_1 ≤ fun a => f a), SimpleFunc.lintegral g_1 (Measure.map g μ) ** rw [← f₀.extend_comp_eq hg (const _ 0), ← SimpleFunc.lintegral_map, ←
SimpleFunc.lintegral_eq_lintegral, ← lintegral] ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β g : α → β hg : MeasurableEmbedding g f : β → ℝ≥0∞ f₀ : α →ₛ ℝ≥0∞ hf₀ : ↑f₀ ≤ fun a => f (g a) ⊢ ∫⁻ (a : β), ↑(SimpleFunc.extend f₀ g hg (const β 0)) a ∂Measure.map g μ ≤ ∫⁻ (a : β), f a ∂Measure.map g μ ** refine' lintegral_mono_ae (hg.ae_map_iff.2 <| eventually_of_forall fun x => _) ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace β g : α → β hg : MeasurableEmbedding g f : β → ℝ≥0∞ f₀ : α →ₛ ℝ≥0∞ hf₀ : ↑f₀ ≤ fun a => f (g a) x : α ⊢ ↑(SimpleFunc.extend f₀ g hg (const β 0)) (g x) ≤ f (g x) ** exact (extend_apply _ _ _ _).trans_le (hf₀ _) ** Qed
| |
MeasureTheory.MeasurePreserving.lintegral_map_equiv ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν✝ : Measure α inst✝ : MeasurableSpace β ν : Measure β f : β → ℝ≥0∞ g : α ≃ᵐ β hg : MeasurePreserving ↑g ⊢ ∫⁻ (a : β), f a ∂ν = ∫⁻ (a : α), f (↑g a) ∂μ ** rw [← MeasureTheory.lintegral_map_equiv f g, hg.map_eq] ** Qed
| |
MeasureTheory.MeasurePreserving.lintegral_comp ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν✝ : Measure α mb : MeasurableSpace β ν : Measure β g : α → β hg : MeasurePreserving g f : β → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂ν ** rw [← hg.map_eq, lintegral_map hf hg.measurable] ** Qed
| |
MeasureTheory.MeasurePreserving.lintegral_comp_emb ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν✝ : Measure α mb : MeasurableSpace β ν : Measure β g : α → β hg : MeasurePreserving g hge : MeasurableEmbedding g f : β → ℝ≥0∞ ⊢ ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂ν ** rw [← hg.map_eq, hge.lintegral_map] ** Qed
| |
MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage_emb ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν✝ : Measure α mb : MeasurableSpace β ν : Measure β g : α → β hg : MeasurePreserving g hge : MeasurableEmbedding g f : β → ℝ≥0∞ s : Set β ⊢ ∫⁻ (a : α) in g ⁻¹' s, f (g a) ∂μ = ∫⁻ (b : β) in s, f b ∂ν ** rw [← hg.map_eq, hge.restrict_map, hge.lintegral_map] ** Qed
| |
MeasureTheory.MeasurePreserving.set_lintegral_comp_emb ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν✝ : Measure α mb : MeasurableSpace β ν : Measure β g : α → β hg : MeasurePreserving g hge : MeasurableEmbedding g f : β → ℝ≥0∞ s : Set α ⊢ ∫⁻ (a : α) in s, f (g a) ∂μ = ∫⁻ (b : β) in g '' s, f b ∂ν ** rw [← hg.set_lintegral_comp_preimage_emb hge, preimage_image_eq _ hge.injective] ** Qed
| |
MeasureTheory.lintegral_dirac' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace α a : α f : α → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), f a ∂dirac a = f a ** simp [lintegral_congr_ae (ae_eq_dirac' hf)] ** Qed
| |
MeasureTheory.lintegral_dirac ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α a : α f : α → ℝ≥0∞ ⊢ ∫⁻ (a : α), f a ∂dirac a = f a ** simp [lintegral_congr_ae (ae_eq_dirac f)] ** Qed
| |
MeasureTheory.set_lintegral_dirac' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α a : α f : α → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) ⊢ ∫⁻ (x : α) in s, f x ∂dirac a = if a ∈ s then f a else 0 ** rw [restrict_dirac' hs] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α a : α f : α → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) ⊢ (∫⁻ (x : α), f x ∂if a ∈ s then dirac a else 0) = if a ∈ s then f a else 0 ** split_ifs ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α a : α f : α → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) h✝ : a ∈ s ⊢ ∫⁻ (x : α), f x ∂dirac a = f a ** exact lintegral_dirac' _ hf ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α a : α f : α → ℝ≥0∞ hf : Measurable f s : Set α hs : MeasurableSet s inst✝ : Decidable (a ∈ s) h✝ : ¬a ∈ s ⊢ ∫⁻ (x : α), f x ∂0 = 0 ** exact lintegral_zero_measure _ ** Qed
| |
MeasureTheory.set_lintegral_dirac ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝² : MeasurableSpace α a : α f : α → ℝ≥0∞ s : Set α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) ⊢ ∫⁻ (x : α) in s, f x ∂dirac a = if a ∈ s then f a else 0 ** rw [restrict_dirac] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝² : MeasurableSpace α a : α f : α → ℝ≥0∞ s : Set α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) ⊢ (∫⁻ (x : α), f x ∂if a ∈ s then dirac a else 0) = if a ∈ s then f a else 0 ** split_ifs ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝² : MeasurableSpace α a : α f : α → ℝ≥0∞ s : Set α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) h✝ : a ∈ s ⊢ ∫⁻ (x : α), f x ∂dirac a = f a ** exact lintegral_dirac _ _ ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝² : MeasurableSpace α a : α f : α → ℝ≥0∞ s : Set α inst✝¹ : MeasurableSingletonClass α inst✝ : Decidable (a ∈ s) h✝ : ¬a ∈ s ⊢ ∫⁻ (x : α), f x ∂0 = 0 ** exact lintegral_zero_measure _ ** Qed
| |
MeasureTheory.lintegral_count' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace α f : α → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), f a ∂count = ∑' (a : α), f a ** rw [count, lintegral_sum_measure] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace α f : α → ℝ≥0∞ hf : Measurable f ⊢ ∑' (i : α), ∫⁻ (a : α), f a ∂dirac i = ∑' (a : α), f a ** congr ** case e_f α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSpace α f : α → ℝ≥0∞ hf : Measurable f ⊢ (fun i => ∫⁻ (a : α), f a ∂dirac i) = fun a => f a ** exact funext fun a => lintegral_dirac' a hf ** Qed
| |
MeasureTheory.lintegral_count ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α f : α → ℝ≥0∞ ⊢ ∫⁻ (a : α), f a ∂count = ∑' (a : α), f a ** rw [count, lintegral_sum_measure] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α f : α → ℝ≥0∞ ⊢ ∑' (i : α), ∫⁻ (a : α), f a ∂dirac i = ∑' (a : α), f a ** congr ** case e_f α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α f : α → ℝ≥0∞ ⊢ (fun i => ∫⁻ (a : α), f a ∂dirac i) = fun a => f a ** exact funext fun a => lintegral_dirac a f ** Qed
| |
ENNReal.tsum_const_eq ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α c : ℝ≥0∞ ⊢ ∑' (x : α), c = c * ↑↑count univ ** rw [← lintegral_count, lintegral_const] ** Qed
| |
ENNReal.count_const_le_le_of_tsum_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α a : α → ℝ≥0∞ a_mble : Measurable a c : ℝ≥0∞ tsum_le_c : ∑' (i : α), a i ≤ c ε : ℝ≥0∞ ε_ne_zero : ε ≠ 0 ε_ne_top : ε ≠ ⊤ ⊢ ↑↑count {i | ε ≤ a i} ≤ c / ε ** rw [← lintegral_count] at tsum_le_c ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α a : α → ℝ≥0∞ a_mble : Measurable a c : ℝ≥0∞ tsum_le_c✝ : ∑' (i : α), a i ≤ c tsum_le_c : ∫⁻ (a_1 : α), a a_1 ∂count ≤ c ε : ℝ≥0∞ ε_ne_zero : ε ≠ 0 ε_ne_top : ε ≠ ⊤ ⊢ ↑↑count {i | ε ≤ a i} ≤ c / ε ** apply (MeasureTheory.meas_ge_le_lintegral_div a_mble.aemeasurable ε_ne_zero ε_ne_top).trans ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : MeasurableSpace α inst✝ : MeasurableSingletonClass α a : α → ℝ≥0∞ a_mble : Measurable a c : ℝ≥0∞ tsum_le_c✝ : ∑' (i : α), a i ≤ c tsum_le_c : ∫⁻ (a_1 : α), a a_1 ∂count ≤ c ε : ℝ≥0∞ ε_ne_zero : ε ≠ 0 ε_ne_top : ε ≠ ⊤ ⊢ (∫⁻ (a_1 : α), a a_1 ∂count) / ε ≤ c / ε ** exact ENNReal.div_le_div tsum_le_c rfl.le ** Qed
| |
MeasureTheory.lintegral_countable' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : Countable α inst✝ : MeasurableSingletonClass α f : α → ℝ≥0∞ ⊢ ∫⁻ (a : α), f a ∂μ = ∑' (a : α), f a * ↑↑μ {a} ** conv_lhs => rw [← sum_smul_dirac μ, lintegral_sum_measure] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : Countable α inst✝ : MeasurableSingletonClass α f : α → ℝ≥0∞ ⊢ ∑' (i : α), ∫⁻ (a : α), f a ∂↑↑μ {i} • dirac i = ∑' (a : α), f a * ↑↑μ {a} ** congr 1 with a : 1 ** case e_f.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝¹ : Countable α inst✝ : MeasurableSingletonClass α f : α → ℝ≥0∞ a : α ⊢ ∫⁻ (a : α), f a ∂↑↑μ {a} • dirac a = f a * ↑↑μ {a} ** rw [lintegral_smul_measure, lintegral_dirac, mul_comm] ** Qed
| |
MeasureTheory.lintegral_singleton' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f a : α ⊢ ∫⁻ (x : α) in {a}, f x ∂μ = f a * ↑↑μ {a} ** simp only [restrict_singleton, lintegral_smul_measure, lintegral_dirac' _ hf, mul_comm] ** Qed
| |
MeasureTheory.lintegral_singleton ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSingletonClass α f : α → ℝ≥0∞ a : α ⊢ ∫⁻ (x : α) in {a}, f x ∂μ = f a * ↑↑μ {a} ** simp only [restrict_singleton, lintegral_smul_measure, lintegral_dirac, mul_comm] ** Qed
| |
MeasureTheory.lintegral_insert ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSingletonClass α a : α s : Set α h : ¬a ∈ s f : α → ℝ≥0∞ ⊢ ∫⁻ (x : α) in insert a s, f x ∂μ = f a * ↑↑μ {a} + ∫⁻ (x : α) in s, f x ∂μ ** rw [← union_singleton, lintegral_union (measurableSet_singleton a), lintegral_singleton,
add_comm] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α inst✝ : MeasurableSingletonClass α a : α s : Set α h : ¬a ∈ s f : α → ℝ≥0∞ ⊢ Disjoint s {a} ** rwa [disjoint_singleton_right] ** Qed
| |
MeasureTheory.ae_lt_top ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, f x < ⊤ ** simp_rw [ae_iff, ENNReal.not_lt_top] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ ⊢ ↑↑μ {a | f a = ⊤} = 0 ** by_contra h ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h : ¬↑↑μ {a | f a = ⊤} = 0 ⊢ False ** apply h2f.lt_top.not_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h : ¬↑↑μ {a | f a = ⊤} = 0 ⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ ** have : (f ⁻¹' {∞}).indicator ⊤ ≤ f := by
intro x
by_cases hx : x ∈ f ⁻¹' {∞} <;> [simpa [indicator_of_mem hx]; simp [indicator_of_not_mem hx]] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h : ¬↑↑μ {a | f a = ⊤} = 0 this : indicator (f ⁻¹' {⊤}) ⊤ ≤ f ⊢ ⊤ ≤ ∫⁻ (x : α), f x ∂μ ** convert lintegral_mono this ** case h.e'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h : ¬↑↑μ {a | f a = ⊤} = 0 this : indicator (f ⁻¹' {⊤}) ⊤ ≤ f ⊢ ⊤ = ∫⁻ (a : α), indicator (f ⁻¹' {⊤}) ⊤ a ∂μ ** rw [lintegral_indicator _ (hf (measurableSet_singleton ∞))] ** case h.e'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h : ¬↑↑μ {a | f a = ⊤} = 0 this : indicator (f ⁻¹' {⊤}) ⊤ ≤ f ⊢ ⊤ = ∫⁻ (a : α) in f ⁻¹' {⊤}, ⊤ a ∂μ ** simp [ENNReal.top_mul', preimage, h] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h : ¬↑↑μ {a | f a = ⊤} = 0 ⊢ indicator (f ⁻¹' {⊤}) ⊤ ≤ f ** intro x ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : Measurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h : ¬↑↑μ {a | f a = ⊤} = 0 x : α ⊢ indicator (f ⁻¹' {⊤}) ⊤ x ≤ f x ** by_cases hx : x ∈ f ⁻¹' {∞} <;> [simpa [indicator_of_mem hx]; simp [indicator_of_not_mem hx]] ** Qed
| |
MeasureTheory.ae_lt_top' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ ⊢ ∫⁻ (x : α), AEMeasurable.mk f hf x ∂μ ≠ ⊤ ** rwa [← lintegral_congr_ae hf.ae_eq_mk] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α f : α → ℝ≥0∞ hf : AEMeasurable f h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤ h2f_meas : ∫⁻ (x : α), AEMeasurable.mk f hf x ∂μ ≠ ⊤ x : α hx : f x = AEMeasurable.mk f hf x h : AEMeasurable.mk f hf x < ⊤ ⊢ f x < ⊤ ** rwa [hx] ** Qed
| |
MeasureTheory.set_lintegral_lt_top_of_bddAbove ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ f : α → ℝ≥0 hf : Measurable f hbdd : BddAbove (f '' s) ⊢ ∫⁻ (x : α) in s, ↑(f x) ∂μ < ⊤ ** obtain ⟨M, hM⟩ := hbdd ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ f : α → ℝ≥0 hf : Measurable f M : ℝ≥0 hM : M ∈ upperBounds (f '' s) ⊢ ∫⁻ (x : α) in s, ↑(f x) ∂μ < ⊤ ** rw [mem_upperBounds] at hM ** case intro α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ f : α → ℝ≥0 hf : Measurable f M : ℝ≥0 hM : ∀ (x : ℝ≥0), x ∈ f '' s → x ≤ M ⊢ ∫⁻ (x : α) in s, ↑(f x) ∂μ < ⊤ ** refine'
lt_of_le_of_lt (set_lintegral_mono hf.coe_nnreal_ennreal (@measurable_const _ _ _ _ ↑M) _) _ ** case intro.refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ f : α → ℝ≥0 hf : Measurable f M : ℝ≥0 hM : ∀ (x : ℝ≥0), x ∈ f '' s → x ≤ M ⊢ ∀ (x : α), x ∈ s → ↑(f x) ≤ ↑M ** simpa using hM ** case intro.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ f : α → ℝ≥0 hf : Measurable f M : ℝ≥0 hM : ∀ (x : ℝ≥0), x ∈ f '' s → x ≤ M ⊢ ∫⁻ (x : α) in s, ↑M ∂μ < ⊤ ** rw [lintegral_const] ** case intro.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ f : α → ℝ≥0 hf : Measurable f M : ℝ≥0 hM : ∀ (x : ℝ≥0), x ∈ f '' s → x ≤ M ⊢ ↑M * ↑↑(Measure.restrict μ s) univ < ⊤ ** refine' ENNReal.mul_lt_top ENNReal.coe_lt_top.ne _ ** case intro.refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α μ ν : Measure α s : Set α hs : ↑↑μ s ≠ ⊤ f : α → ℝ≥0 hf : Measurable f M : ℝ≥0 hM : ∀ (x : ℝ≥0), x ∈ f '' s → x ≤ M ⊢ ↑↑(Measure.restrict μ s) univ ≠ ⊤ ** simp [hs] ** Qed
| |
IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal ** α✝ : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α✝ μ✝ ν : Measure α✝ α : Type u_5 inst✝ : MeasurableSpace α μ : Measure α μ_fin : IsFiniteMeasure μ f : α → ℝ≥0∞ f_bdd : ∃ c, ∀ (x : α), f x ≤ ↑c ⊢ ∫⁻ (x : α), f x ∂μ < ⊤ ** cases' f_bdd with c hc ** case intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α✝ μ✝ ν : Measure α✝ α : Type u_5 inst✝ : MeasurableSpace α μ : Measure α μ_fin : IsFiniteMeasure μ f : α → ℝ≥0∞ c : ℝ≥0 hc : ∀ (x : α), f x ≤ ↑c ⊢ ∫⁻ (x : α), f x ∂μ < ⊤ ** apply lt_of_le_of_lt (@lintegral_mono _ _ μ _ _ hc) ** case intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α✝ μ✝ ν : Measure α✝ α : Type u_5 inst✝ : MeasurableSpace α μ : Measure α μ_fin : IsFiniteMeasure μ f : α → ℝ≥0∞ c : ℝ≥0 hc : ∀ (x : α), f x ≤ ↑c ⊢ ∫⁻ (a : α), ↑c ∂μ < ⊤ ** rw [lintegral_const] ** case intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m : MeasurableSpace α✝ μ✝ ν : Measure α✝ α : Type u_5 inst✝ : MeasurableSpace α μ : Measure α μ_fin : IsFiniteMeasure μ f : α → ℝ≥0∞ c : ℝ≥0 hc : ∀ (x : α), f x ≤ ↑c ⊢ ↑c * ↑↑μ univ < ⊤ ** exact ENNReal.mul_lt_top ENNReal.coe_lt_top.ne μ_fin.measure_univ_lt_top.ne ** Qed
| |
MeasureTheory.lintegral_trim ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ≥0∞ hf : Measurable f ⊢ ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ ** refine'
@Measurable.ennreal_induction α m (fun f => ∫⁻ a, f a ∂μ.trim hm = ∫⁻ a, f a ∂μ) _ _ _ f hf ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ≥0∞ hf : Measurable f ⊢ ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) (indicator s fun x => c) ** intro c s hs ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ≥0∞ hf : Measurable f c : ℝ≥0∞ s : Set α hs : MeasurableSet s ⊢ ∫⁻ (a : α), indicator s (fun x => c) a ∂Measure.trim μ hm = ∫⁻ (a : α), indicator s (fun x => c) a ∂μ ** rw [lintegral_indicator _ hs, lintegral_indicator _ (hm s hs), set_lintegral_const,
set_lintegral_const] ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ≥0∞ hf : Measurable f c : ℝ≥0∞ s : Set α hs : MeasurableSet s ⊢ c * ↑↑(Measure.trim μ hm) s = c * ↑↑μ s ** suffices h_trim_s : μ.trim hm s = μ s ** case h_trim_s α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ≥0∞ hf : Measurable f c : ℝ≥0∞ s : Set α hs : MeasurableSet s ⊢ ↑↑(Measure.trim μ hm) s = ↑↑μ s ** exact trim_measurableSet_eq hm hs ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ≥0∞ hf : Measurable f c : ℝ≥0∞ s : Set α hs : MeasurableSet s h_trim_s : ↑↑(Measure.trim μ hm) s = ↑↑μ s ⊢ c * ↑↑(Measure.trim μ hm) s = c * ↑↑μ s ** rw [h_trim_s] ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ≥0∞ hf : Measurable f ⊢ ∀ ⦃f g : α → ℝ≥0∞⦄, Disjoint (support f) (support g) → Measurable f → Measurable g → (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) f → (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) g → (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) (f + g) ** intro f g _ hf _ hf_prop hg_prop ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ f g : α → ℝ≥0∞ a✝¹ : Disjoint (support f) (support g) hf : Measurable f a✝ : Measurable g hf_prop : ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ hg_prop : ∫⁻ (a : α), g a ∂Measure.trim μ hm = ∫⁻ (a : α), g a ∂μ ⊢ ∫⁻ (a : α), (f + g) a ∂Measure.trim μ hm = ∫⁻ (a : α), (f + g) a ∂μ ** have h_m := lintegral_add_left (μ := Measure.trim μ hm) hf g ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ f g : α → ℝ≥0∞ a✝¹ : Disjoint (support f) (support g) hf : Measurable f a✝ : Measurable g hf_prop : ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ hg_prop : ∫⁻ (a : α), g a ∂Measure.trim μ hm = ∫⁻ (a : α), g a ∂μ h_m : ∫⁻ (a : α), f a + g a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂Measure.trim μ hm + ∫⁻ (a : α), g a ∂Measure.trim μ hm ⊢ ∫⁻ (a : α), (f + g) a ∂Measure.trim μ hm = ∫⁻ (a : α), (f + g) a ∂μ ** have h_m0 := lintegral_add_left (μ := μ) (Measurable.mono hf hm le_rfl) g ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ f g : α → ℝ≥0∞ a✝¹ : Disjoint (support f) (support g) hf : Measurable f a✝ : Measurable g hf_prop : ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ hg_prop : ∫⁻ (a : α), g a ∂Measure.trim μ hm = ∫⁻ (a : α), g a ∂μ h_m : ∫⁻ (a : α), f a + g a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂Measure.trim μ hm + ∫⁻ (a : α), g a ∂Measure.trim μ hm h_m0 : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ ⊢ ∫⁻ (a : α), (f + g) a ∂Measure.trim μ hm = ∫⁻ (a : α), (f + g) a ∂μ ** rwa [hf_prop, hg_prop, ← h_m0] at h_m ** case refine'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ≥0∞ hf : Measurable f ⊢ ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) (f n)) → (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) fun x => ⨆ n, f n x ** intro f hf hf_mono hf_prop ** case refine'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) hf_mono : Monotone f hf_prop : ∀ (n : ℕ), (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) (f n) ⊢ ∫⁻ (a : α), (fun x => ⨆ n, f n x) a ∂Measure.trim μ hm = ∫⁻ (a : α), (fun x => ⨆ n, f n x) a ∂μ ** rw [lintegral_iSup hf hf_mono] ** case refine'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) hf_mono : Monotone f hf_prop : ∀ (n : ℕ), (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) (f n) ⊢ ⨆ n, ∫⁻ (a : α), f n a ∂Measure.trim μ hm = ∫⁻ (a : α), (fun x => ⨆ n, f n x) a ∂μ ** rw [lintegral_iSup (fun n => Measurable.mono (hf n) hm le_rfl) hf_mono] ** case refine'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) hf_mono : Monotone f hf_prop : ∀ (n : ℕ), (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) (f n) ⊢ ⨆ n, ∫⁻ (a : α), f n a ∂Measure.trim μ hm = ⨆ n, ∫⁻ (a : α), f n a ∂μ ** congr ** case refine'_3.e_s α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f✝ : α → ℝ≥0∞ hf✝ : Measurable f✝ f : ℕ → α → ℝ≥0∞ hf : ∀ (n : ℕ), Measurable (f n) hf_mono : Monotone f hf_prop : ∀ (n : ℕ), (fun f => ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ) (f n) ⊢ (fun n => ∫⁻ (a : α), f n a ∂Measure.trim μ hm) = fun n => ∫⁻ (a : α), f n a ∂μ ** exact funext fun n => hf_prop n ** Qed
| |
MeasureTheory.lintegral_trim_ae ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α μ : Measure α hm : m ≤ m0 f : α → ℝ≥0∞ hf : AEMeasurable f ⊢ ∫⁻ (a : α), f a ∂Measure.trim μ hm = ∫⁻ (a : α), f a ∂μ ** rw [lintegral_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk), lintegral_congr_ae hf.ae_eq_mk,
lintegral_trim hm hf.measurable_mk] ** Qed
| |
MeasureTheory.univ_le_of_forall_fin_meas_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α E : Type u_5 inst✝³ : NormedAddCommGroup E inst✝² : MeasurableSpace E inst✝¹ : OpensMeasurableSpace E μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) C : ℝ≥0∞ f : Set α → ℝ≥0∞ hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n) ⊢ f univ ≤ C ** let S := @spanningSets _ m (μ.trim hm) _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α E : Type u_5 inst✝³ : NormedAddCommGroup E inst✝² : MeasurableSpace E inst✝¹ : OpensMeasurableSpace E μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) C : ℝ≥0∞ f : Set α → ℝ≥0∞ hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n) S : ℕ → Set α := spanningSets (Measure.trim μ hm) ⊢ f univ ≤ C ** have hS_mono : Monotone S := @monotone_spanningSets _ m (μ.trim hm) _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α E : Type u_5 inst✝³ : NormedAddCommGroup E inst✝² : MeasurableSpace E inst✝¹ : OpensMeasurableSpace E μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) C : ℝ≥0∞ f : Set α → ℝ≥0∞ hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n) S : ℕ → Set α := spanningSets (Measure.trim μ hm) hS_mono : Monotone S ⊢ f univ ≤ C ** have hS_meas : ∀ n, MeasurableSet[m] (S n) := @measurable_spanningSets _ m (μ.trim hm) _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α E : Type u_5 inst✝³ : NormedAddCommGroup E inst✝² : MeasurableSpace E inst✝¹ : OpensMeasurableSpace E μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) C : ℝ≥0∞ f : Set α → ℝ≥0∞ hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n) S : ℕ → Set α := spanningSets (Measure.trim μ hm) hS_mono : Monotone S hS_meas : ∀ (n : ℕ), MeasurableSet (S n) ⊢ f univ ≤ C ** rw [← @iUnion_spanningSets _ m (μ.trim hm)] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α E : Type u_5 inst✝³ : NormedAddCommGroup E inst✝² : MeasurableSpace E inst✝¹ : OpensMeasurableSpace E μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) C : ℝ≥0∞ f : Set α → ℝ≥0∞ hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n) S : ℕ → Set α := spanningSets (Measure.trim μ hm) hS_mono : Monotone S hS_meas : ∀ (n : ℕ), MeasurableSet (S n) ⊢ f (⋃ i, spanningSets (Measure.trim μ hm) i) ≤ C ** refine' (h_F_lim S hS_meas hS_mono).trans _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α E : Type u_5 inst✝³ : NormedAddCommGroup E inst✝² : MeasurableSpace E inst✝¹ : OpensMeasurableSpace E μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) C : ℝ≥0∞ f : Set α → ℝ≥0∞ hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n) S : ℕ → Set α := spanningSets (Measure.trim μ hm) hS_mono : Monotone S hS_meas : ∀ (n : ℕ), MeasurableSet (S n) ⊢ ⨆ n, f (S n) ≤ C ** refine' iSup_le fun n => hf (S n) (hS_meas n) _ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α E : Type u_5 inst✝³ : NormedAddCommGroup E inst✝² : MeasurableSpace E inst✝¹ : OpensMeasurableSpace E μ : Measure α hm : m ≤ m0 inst✝ : SigmaFinite (Measure.trim μ hm) C : ℝ≥0∞ f : Set α → ℝ≥0∞ hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n) S : ℕ → Set α := spanningSets (Measure.trim μ hm) hS_mono : Monotone S hS_meas : ∀ (n : ℕ), MeasurableSet (S n) n : ℕ ⊢ ↑↑μ (S n) ≠ ⊤ ** exact ((le_trim hm).trans_lt (@measure_spanningSets_lt_top _ m (μ.trim hm) _ n)).ne ** Qed
| |
MeasureTheory.lintegral_le_of_forall_fin_meas_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 m m0 : MeasurableSpace α E : Type u_5 inst✝⁴ : NormedAddCommGroup E inst✝³ : MeasurableSpace E inst✝² : OpensMeasurableSpace E inst✝¹ : MeasurableSpace α μ : Measure α inst✝ : SigmaFinite μ C : ℝ≥0∞ f : α → ℝ≥0∞ hf_meas : AEMeasurable f hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C ⊢ SigmaFinite (Measure.trim μ (_ : inst✝¹ ≤ inst✝¹)) ** rwa [trim_eq_self] ** Qed
| |
MeasureTheory.OuterMeasure.IsMetric.le_caratheodory ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y μ : OuterMeasure X inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X hm : IsMetric μ ⊢ inst✝¹ ≤ OuterMeasure.caratheodory μ ** rw [BorelSpace.measurable_eq (α := X)] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y μ : OuterMeasure X inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X hm : IsMetric μ ⊢ borel X ≤ OuterMeasure.caratheodory μ ** exact hm.borel_le_caratheodory ** Qed
| |
MeasureTheory.OuterMeasure.mkMetric'.le_pre ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m : Set X → ℝ≥0∞ r : ℝ≥0∞ μ : OuterMeasure X s : Set X ⊢ μ ≤ pre m r ↔ ∀ (s : Set X), diam s ≤ r → ↑μ s ≤ m s ** simp only [pre, le_boundedBy, extend, le_iInf_iff] ** Qed
| |
MeasureTheory.OuterMeasure.mkMetric'.mono_pre_nat ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m✝ : Set X → ℝ≥0∞ r : ℝ≥0∞ μ : OuterMeasure X s✝ : Set X m : Set X → ℝ≥0∞ k l : ℕ h : k ≤ l s : Set X hs : diam s ≤ (↑l)⁻¹ ⊢ (↑l)⁻¹ ≤ (↑k)⁻¹ ** simpa ** Qed
| |
MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre_nat ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m✝ : Set X → ℝ≥0∞ r : ℝ≥0∞ μ : OuterMeasure X s✝ : Set X m : Set X → ℝ≥0∞ s : Set X ⊢ Tendsto (fun n => ↑(pre m (↑n)⁻¹) s) atTop (𝓝 (↑(mkMetric' m) s)) ** refine' (tendsto_pre m s).comp (tendsto_inf.2 ⟨ENNReal.tendsto_inv_nat_nhds_zero, _⟩) ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m✝ : Set X → ℝ≥0∞ r : ℝ≥0∞ μ : OuterMeasure X s✝ : Set X m : Set X → ℝ≥0∞ s : Set X ⊢ Tendsto (fun n => (↑n)⁻¹) atTop (𝓟 (Ioi 0)) ** refine' tendsto_principal.2 (eventually_of_forall fun n => _) ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m✝ : Set X → ℝ≥0∞ r : ℝ≥0∞ μ : OuterMeasure X s✝ : Set X m : Set X → ℝ≥0∞ s : Set X n : ℕ ⊢ (↑n)⁻¹ ∈ Ioi 0 ** simp ** Qed
| |
MeasureTheory.OuterMeasure.mkMetric_mono_smul ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ ⊢ mkMetric m₁ ≤ c • mkMetric m₂ ** rcases (mem_nhdsWithin_Ici_iff_exists_Ico_subset' zero_lt_one).1 hle with ⟨r, hr0, hr⟩ ** case intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ r : ℝ≥0∞ hr0 : r ∈ Ioi 0 hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x} ⊢ mkMetric m₁ ≤ c • mkMetric m₂ ** refine' fun s =>
le_of_tendsto_of_tendsto (mkMetric'.tendsto_pre _ s)
(ENNReal.Tendsto.const_mul (mkMetric'.tendsto_pre _ s) (Or.inr hc))
(mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, hr0⟩) fun r' hr' => _) ** case intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ r : ℝ≥0∞ hr0 : r ∈ Ioi 0 hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x} s : Set X r' : ℝ≥0∞ hr' : r' ∈ Ioo 0 r ⊢ r' ∈ {x | (fun x => (fun r => ↑(mkMetric'.pre (fun s => m₁ (diam s)) r) s) x ≤ (fun b => ↑(RingHom.id ℝ≥0∞) c * ↑(mkMetric'.pre (fun s => m₂ (diam s)) b) s) x) x} ** simp only [mem_setOf_eq, mkMetric'.pre, RingHom.id_apply] ** case intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ r : ℝ≥0∞ hr0 : r ∈ Ioi 0 hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x} s : Set X r' : ℝ≥0∞ hr' : r' ∈ Ioo 0 r ⊢ ↑(boundedBy (extend fun s x => m₁ (diam s))) s ≤ c * ↑(boundedBy (extend fun s x => m₂ (diam s))) s ** rw [← smul_eq_mul, ← smul_apply, smul_boundedBy hc] ** case intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ r : ℝ≥0∞ hr0 : r ∈ Ioi 0 hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x} s : Set X r' : ℝ≥0∞ hr' : r' ∈ Ioo 0 r ⊢ ↑(boundedBy (extend fun s x => m₁ (diam s))) s ≤ ↑(boundedBy (c • extend fun s x => m₂ (diam s))) s ** refine' le_boundedBy.2 (fun t => (boundedBy_le _).trans _) _ ** case intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ r : ℝ≥0∞ hr0 : r ∈ Ioi 0 hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x} s : Set X r' : ℝ≥0∞ hr' : r' ∈ Ioo 0 r t : Set X ⊢ extend (fun s x => m₁ (diam s)) t ≤ (c • extend fun s x => m₂ (diam s)) t ** simp only [smul_eq_mul, Pi.smul_apply, extend, iInf_eq_if] ** case intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ r : ℝ≥0∞ hr0 : r ∈ Ioi 0 hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x} s : Set X r' : ℝ≥0∞ hr' : r' ∈ Ioo 0 r t : Set X ⊢ (if diam t ≤ r' then m₁ (diam t) else ⊤) ≤ c * if diam t ≤ r' then m₂ (diam t) else ⊤ ** split_ifs with ht ** case pos ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ r : ℝ≥0∞ hr0 : r ∈ Ioi 0 hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x} s : Set X r' : ℝ≥0∞ hr' : r' ∈ Ioo 0 r t : Set X ht : diam t ≤ r' ⊢ m₁ (diam t) ≤ c * m₂ (diam t) ** apply hr ** case pos.a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ r : ℝ≥0∞ hr0 : r ∈ Ioi 0 hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x} s : Set X r' : ℝ≥0∞ hr' : r' ∈ Ioo 0 r t : Set X ht : diam t ≤ r' ⊢ diam t ∈ Ico 0 r ** exact ⟨zero_le _, ht.trans_lt hr'.2⟩ ** case neg ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ r : ℝ≥0∞ hr0 : r ∈ Ioi 0 hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x} s : Set X r' : ℝ≥0∞ hr' : r' ∈ Ioo 0 r t : Set X ht : ¬diam t ≤ r' ⊢ ⊤ ≤ c * ⊤ ** simp [h0] ** Qed
| |
MeasureTheory.OuterMeasure.mkMetric_mono ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] m₂ ⊢ mkMetric m₁ ≤ mkMetric m₂ ** convert @mkMetric_mono_smul X _ _ m₂ _ ENNReal.one_ne_top one_ne_zero _ <;> simp [*] ** Qed
| |
MeasureTheory.OuterMeasure.isometry_map_mkMetric ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹ : EMetricSpace X inst✝ : EMetricSpace Y m : ℝ≥0∞ → ℝ≥0∞ f : X → Y hf : Isometry f H : Monotone m ∨ Surjective f ⊢ ↑(map f) (mkMetric m) = ↑(restrict (range f)) (mkMetric m) ** rw [← isometry_comap_mkMetric _ hf H, map_comap] ** Qed
| |
MeasureTheory.OuterMeasure.coe_mkMetric ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ ⊢ ↑(mkMetric m) = ↑↑(Measure.mkMetric m) ** rw [← Measure.mkMetric_toOuterMeasure] ** Qed
| |
MeasureTheory.Measure.mkMetric_mono_smul ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ ⊢ mkMetric m₁ ≤ c • mkMetric m₂ ** intro s _ ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ s : Set X a✝ : MeasurableSet s ⊢ ↑↑(mkMetric m₁) s ≤ ↑↑(c • mkMetric m₂) s ** rw [← OuterMeasure.coe_mkMetric, coe_smul, ← OuterMeasure.coe_mkMetric] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ c : ℝ≥0∞ hc : c ≠ ⊤ h0 : c ≠ 0 hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] c • m₂ s : Set X a✝ : MeasurableSet s ⊢ ↑(OuterMeasure.mkMetric m₁) s ≤ (c • ↑(OuterMeasure.mkMetric m₂)) s ** exact OuterMeasure.mkMetric_mono_smul hc h0 hle s ** Qed
| |
MeasureTheory.Measure.mkMetric_mono ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m₁ m₂ : ℝ≥0∞ → ℝ≥0∞ hle : m₁ ≤ᶠ[𝓝[Ici 0] 0] m₂ ⊢ mkMetric m₁ ≤ mkMetric m₂ ** convert @mkMetric_mono_smul X _ _ _ _ m₂ _ ENNReal.one_ne_top one_ne_zero _ <;> simp [*] ** Qed
| |
MeasureTheory.Measure.mkMetric_apply ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X ⊢ ↑↑(mkMetric m) s = ⨆ r, ⨆ (_ : 0 < r), ⨅ t, ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), m (diam (t n)) ** simp only [← OuterMeasure.coe_mkMetric, OuterMeasure.mkMetric, OuterMeasure.mkMetric',
OuterMeasure.iSup_apply, OuterMeasure.mkMetric'.pre, OuterMeasure.boundedBy_apply, extend] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X ⊢ ⨆ i, ⨆ (_ : i > 0), ⨅ t, ⨅ (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ i), m (diam (t n)) = ⨆ r, ⨆ (_ : 0 < r), ⨅ t, ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), m (diam (t n)) ** refine'
surjective_id.iSup_congr (fun r => r) fun r =>
iSup_congr_Prop Iff.rfl fun _ =>
surjective_id.iInf_congr _ fun t => iInf_congr_Prop Iff.rfl fun ht => _ ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (id t n)), ⨅ (_ : diam (id t n) ≤ r), m (diam (id t n)) = ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ (fun r => r) r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), m (diam (t n)) ** dsimp ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ r), m (diam (t n)) = ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), m (diam (t n)) ** by_cases htr : ∀ n, diam (t n) ≤ r ** case pos ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t htr : ∀ (n : ℕ), diam (t n) ≤ r ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ r), m (diam (t n)) = ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), m (diam (t n)) ** rw [iInf_eq_if, if_pos htr] ** case pos ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t htr : ∀ (n : ℕ), diam (t n) ≤ r ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ r), m (diam (t n)) = ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), m (diam (t n)) ** congr 1 with n : 1 ** case pos.e_f.h ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t htr : ∀ (n : ℕ), diam (t n) ≤ r n : ℕ ⊢ ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ r), m (diam (t n)) = ⨆ (_ : Set.Nonempty (t n)), m (diam (t n)) ** simp only [iInf_eq_if, htr n, id, if_true, iSup_and'] ** case neg ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t htr : ¬∀ (n : ℕ), diam (t n) ≤ r ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ r), m (diam (t n)) = ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), m (diam (t n)) ** rw [iInf_eq_if, if_neg htr] ** case neg ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t htr : ¬∀ (n : ℕ), diam (t n) ≤ r ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ r), m (diam (t n)) = ⊤ ** push_neg at htr ** case neg ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t htr : ∃ n, r < diam (t n) ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ r), m (diam (t n)) = ⊤ ** rcases htr with ⟨n, hn⟩ ** case neg.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t n : ℕ hn : r < diam (t n) ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ r), m (diam (t n)) = ⊤ ** refine' ENNReal.tsum_eq_top_of_eq_top ⟨n, _⟩ ** case neg.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t n : ℕ hn : r < diam (t n) ⊢ ⨆ (_ : Set.Nonempty (t n)), ⨅ (_ : diam (t n) ≤ r), m (diam (t n)) = ⊤ ** rw [iSup_eq_if, if_pos, iInf_eq_if, if_neg] ** case neg.intro.hnc ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t n : ℕ hn : r < diam (t n) ⊢ ¬diam (t n) ≤ r case neg.intro.hc ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t n : ℕ hn : r < diam (t n) ⊢ Set.Nonempty (t n) ** exact hn.not_le ** case neg.intro.hc ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t n : ℕ hn : r < diam (t n) ⊢ Set.Nonempty (t n) ** rcases diam_pos_iff.1 ((zero_le r).trans_lt hn) with ⟨x, hx, -⟩ ** case neg.intro.hc.intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X m : ℝ≥0∞ → ℝ≥0∞ s : Set X r : ℝ≥0∞ x✝ : r > 0 t : ℕ → Set X ht : s ⊆ iUnion t n : ℕ hn : r < diam (t n) x : X hx : x ∈ t n ⊢ Set.Nonempty (t n) ** exact ⟨x, hx⟩ ** Qed
| |
MeasureTheory.Measure.mkMetric_le_liminf_sum ** ι✝ : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X β : Type u_4 ι : β → Type u_5 hι : (n : β) → Fintype (ι n) s : Set X l : Filter β r : β → ℝ≥0∞ hr : Tendsto r l (𝓝 0) t : (n : β) → ι n → Set X ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), diam (t n i) ≤ r n hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ i, t n i m : ℝ≥0∞ → ℝ≥0∞ ⊢ ↑↑(mkMetric m) s ≤ liminf (fun n => ∑ i : ι n, m (diam (t n i))) l ** simpa only [tsum_fintype] using mkMetric_le_liminf_tsum s r hr t ht hst m ** Qed
| |
MeasureTheory.Measure.hausdorffMeasure_zero_or_top ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X ⊢ ↑↑μH[d₂] s = 0 ∨ ↑↑μH[d₁] s = ⊤ ** by_contra' H ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ ⊢ False ** suffices ∀ c : ℝ≥0, c ≠ 0 → μH[d₂] s ≤ c * μH[d₁] s by
rcases ENNReal.exists_nnreal_pos_mul_lt H.2 H.1 with ⟨c, hc0, hc⟩
exact hc.not_le (this c (pos_iff_ne_zero.1 hc0)) ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ ⊢ ∀ (c : ℝ≥0), c ≠ 0 → ↑↑μH[d₂] s ≤ ↑c * ↑↑μH[d₁] s ** intro c hc ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 ⊢ ↑↑μH[d₂] s ≤ ↑c * ↑↑μH[d₁] s ** refine' le_iff'.1 (mkMetric_mono_smul ENNReal.coe_ne_top (by exact_mod_cast hc) _) s ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 ⊢ (fun r => r ^ d₂) ≤ᶠ[𝓝[Ici 0] 0] ↑c • fun r => r ^ d₁ ** have : 0 < ((c : ℝ≥0∞) ^ (d₂ - d₁)⁻¹) := by
rw [ENNReal.coe_rpow_of_ne_zero hc, pos_iff_ne_zero, Ne.def, ENNReal.coe_eq_zero,
NNReal.rpow_eq_zero_iff]
exact mt And.left hc ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ ⊢ (fun r => r ^ d₂) ≤ᶠ[𝓝[Ici 0] 0] ↑c • fun r => r ^ d₁ ** filter_upwards [Ico_mem_nhdsWithin_Ici ⟨le_rfl, this⟩] ** case h ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ ⊢ ∀ (a : ℝ≥0∞), a ∈ Ico 0 (↑c ^ (d₂ - d₁)⁻¹) → a ^ d₂ ≤ (↑c • fun r => r ^ d₁) a ** rintro r ⟨hr₀, hrc⟩ ** case h.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ r : ℝ≥0∞ hr₀ : 0 ≤ r hrc : r < ↑c ^ (d₂ - d₁)⁻¹ ⊢ r ^ d₂ ≤ (↑c • fun r => r ^ d₁) r ** lift r to ℝ≥0 using ne_top_of_lt hrc ** case h.intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ r : ℝ≥0 hr₀ : 0 ≤ ↑r hrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹ ⊢ ↑r ^ d₂ ≤ (↑c • fun r => r ^ d₁) ↑r ** rw [Pi.smul_apply, smul_eq_mul,
← ENNReal.div_le_iff_le_mul (Or.inr ENNReal.coe_ne_top) (Or.inr <| mt ENNReal.coe_eq_zero.1 hc)] ** case h.intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ r : ℝ≥0 hr₀ : 0 ≤ ↑r hrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹ ⊢ ↑r ^ d₂ / ↑r ^ d₁ ≤ ↑c ** rcases eq_or_ne r 0 with (rfl | hr₀) ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ this : ∀ (c : ℝ≥0), c ≠ 0 → ↑↑μH[d₂] s ≤ ↑c * ↑↑μH[d₁] s ⊢ False ** rcases ENNReal.exists_nnreal_pos_mul_lt H.2 H.1 with ⟨c, hc0, hc⟩ ** case intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ this : ∀ (c : ℝ≥0), c ≠ 0 → ↑↑μH[d₂] s ≤ ↑c * ↑↑μH[d₁] s c : ℝ≥0 hc0 : c > 0 hc : ↑c * ↑↑μH[d₁] s < ↑↑μH[d₂] s ⊢ False ** exact hc.not_le (this c (pos_iff_ne_zero.1 hc0)) ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 ⊢ ↑c ≠ 0 ** exact_mod_cast hc ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 ⊢ 0 < ↑c ^ (d₂ - d₁)⁻¹ ** rw [ENNReal.coe_rpow_of_ne_zero hc, pos_iff_ne_zero, Ne.def, ENNReal.coe_eq_zero,
NNReal.rpow_eq_zero_iff] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 ⊢ ¬(c = 0 ∧ (d₂ - d₁)⁻¹ ≠ 0) ** exact mt And.left hc ** case h.intro.intro.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ hr₀ : 0 ≤ ↑0 hrc : ↑0 < ↑c ^ (d₂ - d₁)⁻¹ ⊢ ↑0 ^ d₂ / ↑0 ^ d₁ ≤ ↑c ** rcases lt_or_le 0 d₂ with (h₂ | h₂) ** case h.intro.intro.inl.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ hr₀ : 0 ≤ ↑0 hrc : ↑0 < ↑c ^ (d₂ - d₁)⁻¹ h₂ : 0 < d₂ ⊢ ↑0 ^ d₂ / ↑0 ^ d₁ ≤ ↑c ** simp only [h₂, ENNReal.zero_rpow_of_pos, zero_le, ENNReal.zero_div, ENNReal.coe_zero] ** case h.intro.intro.inl.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ hr₀ : 0 ≤ ↑0 hrc : ↑0 < ↑c ^ (d₂ - d₁)⁻¹ h₂ : d₂ ≤ 0 ⊢ ↑0 ^ d₂ / ↑0 ^ d₁ ≤ ↑c ** simp only [h.trans_le h₂, ENNReal.div_top, zero_le, ENNReal.zero_rpow_of_neg,
ENNReal.coe_zero] ** case h.intro.intro.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ r : ℝ≥0 hr₀✝ : 0 ≤ ↑r hrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹ hr₀ : r ≠ 0 ⊢ ↑r ^ d₂ / ↑r ^ d₁ ≤ ↑c ** have : (r : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using hr₀ ** case h.intro.intro.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this✝ : 0 < ↑c ^ (d₂ - d₁)⁻¹ r : ℝ≥0 hr₀✝ : 0 ≤ ↑r hrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹ hr₀ : r ≠ 0 this : ↑r ≠ 0 ⊢ ↑r ^ d₂ / ↑r ^ d₁ ≤ ↑c ** rw [← ENNReal.rpow_sub _ _ this ENNReal.coe_ne_top] ** case h.intro.intro.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this✝ : 0 < ↑c ^ (d₂ - d₁)⁻¹ r : ℝ≥0 hr₀✝ : 0 ≤ ↑r hrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹ hr₀ : r ≠ 0 this : ↑r ≠ 0 ⊢ ↑r ^ (d₂ - d₁) ≤ ↑c ** refine' (ENNReal.rpow_lt_rpow hrc (sub_pos.2 h)).le.trans _ ** case h.intro.intro.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this✝ : 0 < ↑c ^ (d₂ - d₁)⁻¹ r : ℝ≥0 hr₀✝ : 0 ≤ ↑r hrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹ hr₀ : r ≠ 0 this : ↑r ≠ 0 ⊢ (↑c ^ (d₂ - d₁)⁻¹) ^ (d₂ - d₁) ≤ ↑c ** rw [← ENNReal.rpow_mul, inv_mul_cancel (sub_pos.2 h).ne', ENNReal.rpow_one] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ < d₂ s : Set X H : ↑↑μH[d₂] s ≠ 0 ∧ ↑↑μH[d₁] s ≠ ⊤ c : ℝ≥0 hc : c ≠ 0 this : 0 < ↑c ^ (d₂ - d₁)⁻¹ r : ℝ≥0 hr₀✝ : 0 ≤ ↑r hrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹ hr₀ : r ≠ 0 ⊢ ↑r ≠ 0 ** simpa only [ENNReal.coe_eq_zero, Ne.def] using hr₀ ** Qed
| |
MeasureTheory.Measure.hausdorffMeasure_mono ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h : d₁ ≤ d₂ s : Set X ⊢ ↑↑μH[d₂] s ≤ ↑↑μH[d₁] s ** rcases h.eq_or_lt with (rfl | h) ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h✝ : d₁ ≤ d₂ s : Set X h : d₁ < d₂ ⊢ ↑↑μH[d₂] s ≤ ↑↑μH[d₁] s ** cases' hausdorffMeasure_zero_or_top h s with hs hs ** case inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ : ℝ s : Set X h : d₁ ≤ d₁ ⊢ ↑↑μH[d₁] s ≤ ↑↑μH[d₁] s ** exact le_rfl ** case inr.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h✝ : d₁ ≤ d₂ s : Set X h : d₁ < d₂ hs : ↑↑μH[d₂] s = 0 ⊢ ↑↑μH[d₂] s ≤ ↑↑μH[d₁] s ** rw [hs] ** case inr.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h✝ : d₁ ≤ d₂ s : Set X h : d₁ < d₂ hs : ↑↑μH[d₂] s = 0 ⊢ 0 ≤ ↑↑μH[d₁] s ** exact zero_le _ ** case inr.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h✝ : d₁ ≤ d₂ s : Set X h : d₁ < d₂ hs : ↑↑μH[d₁] s = ⊤ ⊢ ↑↑μH[d₂] s ≤ ↑↑μH[d₁] s ** rw [hs] ** case inr.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d₁ d₂ : ℝ h✝ : d₁ ≤ d₂ s : Set X h : d₁ < d₂ hs : ↑↑μH[d₁] s = ⊤ ⊢ ↑↑μH[d₂] s ≤ ⊤ ** exact le_top ** Qed
| |
MeasureTheory.Measure.noAtoms_hausdorff ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d : ℝ hd : 0 < d ⊢ NoAtoms μH[d] ** refine' ⟨fun x => _⟩ ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d : ℝ hd : 0 < d x : X ⊢ ↑↑μH[d] {x} = 0 ** rw [← nonpos_iff_eq_zero, hausdorffMeasure_apply] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d : ℝ hd : 0 < d x : X ⊢ ⨆ r, ⨆ (_ : 0 < r), ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ d ≤ 0 ** refine' iSup₂_le fun ε _ => iInf₂_le_of_le (fun _ => {x}) _ <| iInf_le_of_le (fun _ => _) _ ** case refine'_1 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d : ℝ hd : 0 < d x : X ε : ℝ≥0∞ x✝ : 0 < ε ⊢ {x} ⊆ ⋃ n, (fun x_1 => {x}) n ** exact subset_iUnion (fun _ => {x} : ℕ → Set X) 0 ** case refine'_2 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d : ℝ hd : 0 < d x : X ε : ℝ≥0∞ x✝¹ : 0 < ε x✝ : ℕ ⊢ diam ((fun x_1 => {x}) x✝) ≤ ε ** simp only [EMetric.diam_singleton, zero_le] ** case refine'_3 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d : ℝ hd : 0 < d x : X ε : ℝ≥0∞ x✝ : 0 < ε ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty ((fun x_1 => {x}) n)), diam ((fun x_2 => {x}) n) ^ d ≤ 0 ** simp [hd] ** Qed
| |
MeasureTheory.Measure.hausdorffMeasure_zero_singleton ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X ⊢ ↑↑μH[0] {x} = 1 ** apply le_antisymm ** case a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X ⊢ ↑↑μH[0] {x} ≤ 1 ** let r : ℕ → ℝ≥0∞ := fun _ => 0 ** case a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X r : ℕ → ℝ≥0∞ := fun x => 0 ⊢ ↑↑μH[0] {x} ≤ 1 ** let t : ℕ → Unit → Set X := fun _ _ => {x} ** case a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X r : ℕ → ℝ≥0∞ := fun x => 0 t : ℕ → Unit → Set X := fun x_1 x_2 => {x} ⊢ ↑↑μH[0] {x} ≤ 1 ** have ht : ∀ᶠ n in atTop, ∀ i, diam (t n i) ≤ r n := by
simp only [imp_true_iff, eq_self_iff_true, diam_singleton, eventually_atTop,
nonpos_iff_eq_zero, exists_const] ** case a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X r : ℕ → ℝ≥0∞ := fun x => 0 t : ℕ → Unit → Set X := fun x_1 x_2 => {x} ht : ∀ᶠ (n : ℕ) in atTop, ∀ (i : Unit), diam (t n i) ≤ r n ⊢ ↑↑μH[0] {x} ≤ 1 ** simpa [liminf_const] using hausdorffMeasure_le_liminf_sum 0 {x} r tendsto_const_nhds t ht ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X r : ℕ → ℝ≥0∞ := fun x => 0 t : ℕ → Unit → Set X := fun x_1 x_2 => {x} ⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : Unit), diam (t n i) ≤ r n ** simp only [imp_true_iff, eq_self_iff_true, diam_singleton, eventually_atTop,
nonpos_iff_eq_zero, exists_const] ** case a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X ⊢ 1 ≤ ↑↑μH[0] {x} ** rw [hausdorffMeasure_apply] ** case a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X ⊢ 1 ≤ ⨆ r, ⨆ (_ : 0 < r), ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 ** suffices
(1 : ℝ≥0∞) ≤
⨅ (t : ℕ → Set X) (_ : {x} ⊆ ⋃ n, t n) (_ : ∀ n, diam (t n) ≤ 1),
∑' n, ⨆ _ : (t n).Nonempty, diam (t n) ^ (0 : ℝ) by
apply le_trans this _
convert le_iSup₂ (α := ℝ≥0∞) (1 : ℝ≥0∞) zero_lt_one
rfl ** case a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X ⊢ 1 ≤ ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 ** simp only [ENNReal.rpow_zero, le_iInf_iff] ** case a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X ⊢ ∀ (i : ℕ → Set X), {x} ⊆ ⋃ n, i n → (∀ (n : ℕ), diam (i n) ≤ 1) → 1 ≤ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (i n)), 1 ** intro t hst _ ** case a ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X t : ℕ → Set X hst : {x} ⊆ ⋃ n, t n i✝ : ∀ (n : ℕ), diam (t n) ≤ 1 ⊢ 1 ≤ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), 1 ** rcases mem_iUnion.1 (hst (mem_singleton x)) with ⟨m, hm⟩ ** case a.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X t : ℕ → Set X hst : {x} ⊆ ⋃ n, t n i✝ : ∀ (n : ℕ), diam (t n) ≤ 1 m : ℕ hm : x ∈ t m ⊢ 1 ≤ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), 1 ** have A : (t m).Nonempty := ⟨x, hm⟩ ** case a.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X t : ℕ → Set X hst : {x} ⊆ ⋃ n, t n i✝ : ∀ (n : ℕ), diam (t n) ≤ 1 m : ℕ hm : x ∈ t m A : Set.Nonempty (t m) ⊢ 1 ≤ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), 1 ** calc
(1 : ℝ≥0∞) = ⨆ h : (t m).Nonempty, 1 := by simp only [A, ciSup_pos]
_ ≤ ∑' n, ⨆ h : (t n).Nonempty, 1 := ENNReal.le_tsum _ ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X this : 1 ≤ ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 ⊢ 1 ≤ ⨆ r, ⨆ (_ : 0 < r), ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 ** apply le_trans this _ ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X this : 1 ≤ ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 ⊢ ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 ≤ ⨆ r, ⨆ (_ : 0 < r), ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 ** convert le_iSup₂ (α := ℝ≥0∞) (1 : ℝ≥0∞) zero_lt_one ** case h.e'_3 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X this : 1 ≤ ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 ⊢ ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 = ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ 0 ** rfl ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X x : X t : ℕ → Set X hst : {x} ⊆ ⋃ n, t n i✝ : ∀ (n : ℕ), diam (t n) ≤ 1 m : ℕ hm : x ∈ t m A : Set.Nonempty (t m) ⊢ 1 = ⨆ (_ : Set.Nonempty (t m)), 1 ** simp only [A, ciSup_pos] ** Qed
| |
MeasureTheory.Measure.one_le_hausdorffMeasure_zero_of_nonempty ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X s : Set X h : Set.Nonempty s ⊢ 1 ≤ ↑↑μH[0] s ** rcases h with ⟨x, hx⟩ ** case intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X s : Set X x : X hx : x ∈ s ⊢ 1 ≤ ↑↑μH[0] s ** calc
(1 : ℝ≥0∞) = μH[0] ({x} : Set X) := (hausdorffMeasure_zero_singleton x).symm
_ ≤ μH[0] s := measure_mono (singleton_subset_iff.2 hx) ** Qed
| |
MeasureTheory.Measure.hausdorffMeasure_le_one_of_subsingleton ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X s : Set X hs : Set.Subsingleton s d : ℝ hd : 0 ≤ d ⊢ ↑↑μH[d] s ≤ 1 ** rcases eq_empty_or_nonempty s with (rfl | ⟨x, hx⟩) ** case inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X d : ℝ hd : 0 ≤ d hs : Set.Subsingleton ∅ ⊢ ↑↑μH[d] ∅ ≤ 1 ** simp only [measure_empty, zero_le] ** case inr.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X s : Set X hs : Set.Subsingleton s d : ℝ hd : 0 ≤ d x : X hx : x ∈ s ⊢ ↑↑μH[d] s ≤ 1 ** rw [(subsingleton_iff_singleton hx).1 hs] ** case inr.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X s : Set X hs : Set.Subsingleton s d : ℝ hd : 0 ≤ d x : X hx : x ∈ s ⊢ ↑↑μH[d] {x} ≤ 1 ** rcases eq_or_lt_of_le hd with (rfl | dpos) ** case inr.intro.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X s : Set X hs : Set.Subsingleton s x : X hx : x ∈ s hd : 0 ≤ 0 ⊢ ↑↑μH[0] {x} ≤ 1 ** simp only [le_refl, hausdorffMeasure_zero_singleton] ** case inr.intro.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X s : Set X hs : Set.Subsingleton s d : ℝ hd : 0 ≤ d x : X hx : x ∈ s dpos : 0 < d ⊢ ↑↑μH[d] {x} ≤ 1 ** haveI := noAtoms_hausdorff X dpos ** case inr.intro.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝³ : EMetricSpace X inst✝² : EMetricSpace Y inst✝¹ : MeasurableSpace X inst✝ : BorelSpace X s : Set X hs : Set.Subsingleton s d : ℝ hd : 0 ≤ d x : X hx : x ∈ s dpos : 0 < d this : NoAtoms μH[d] ⊢ ↑↑μH[d] {x} ≤ 1 ** simp only [zero_le, measure_singleton] ** Qed
| |
HolderOnWith.hausdorffMeasure_image_le ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d ⊢ ↑↑μH[d] (f '' s) ≤ ↑C ^ d * ↑↑μH[↑r * d] s ** rcases (zero_le C).eq_or_lt with (rfl | hC0) ** case inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r d : ℝ hd : 0 ≤ d h : HolderOnWith 0 r f s ⊢ ↑↑μH[d] (f '' s) ≤ ↑0 ^ d * ↑↑μH[↑r * d] s ** rcases eq_empty_or_nonempty s with (rfl | ⟨x, hx⟩) ** case inl.inr.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r d : ℝ hd : 0 ≤ d h : HolderOnWith 0 r f s x : X hx : x ∈ s ⊢ ↑↑μH[d] (f '' s) ≤ ↑0 ^ d * ↑↑μH[↑r * d] s ** have : f '' s = {f x} :=
haveI : (f '' s).Subsingleton := by simpa [diam_eq_zero_iff] using h.ediam_image_le
(subsingleton_iff_singleton (mem_image_of_mem f hx)).1 this ** case inl.inr.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r d : ℝ hd : 0 ≤ d h : HolderOnWith 0 r f s x : X hx : x ∈ s this : f '' s = {f x} ⊢ ↑↑μH[d] (f '' s) ≤ ↑0 ^ d * ↑↑μH[↑r * d] s ** rw [this] ** case inl.inr.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r d : ℝ hd : 0 ≤ d h : HolderOnWith 0 r f s x : X hx : x ∈ s this : f '' s = {f x} ⊢ ↑↑μH[d] {f x} ≤ ↑0 ^ d * ↑↑μH[↑r * d] s ** rcases eq_or_lt_of_le hd with (rfl | h'd) ** case inl.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y t : Set X hr : 0 < r d : ℝ hd : 0 ≤ d h : HolderOnWith 0 r f ∅ ⊢ ↑↑μH[d] (f '' ∅) ≤ ↑0 ^ d * ↑↑μH[↑r * d] ∅ ** simp only [measure_empty, nonpos_iff_eq_zero, mul_zero, image_empty] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r d : ℝ hd : 0 ≤ d h : HolderOnWith 0 r f s x : X hx : x ∈ s ⊢ Set.Subsingleton (f '' s) ** simpa [diam_eq_zero_iff] using h.ediam_image_le ** case inl.inr.intro.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r h : HolderOnWith 0 r f s x : X hx : x ∈ s this : f '' s = {f x} hd : 0 ≤ 0 ⊢ ↑↑μH[0] {f x} ≤ ↑0 ^ 0 * ↑↑μH[↑r * 0] s ** simp only [ENNReal.rpow_zero, one_mul, mul_zero] ** case inl.inr.intro.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r h : HolderOnWith 0 r f s x : X hx : x ∈ s this : f '' s = {f x} hd : 0 ≤ 0 ⊢ ↑↑μH[0] {f x} ≤ ↑↑μH[0] s ** rw [hausdorffMeasure_zero_singleton] ** case inl.inr.intro.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r h : HolderOnWith 0 r f s x : X hx : x ∈ s this : f '' s = {f x} hd : 0 ≤ 0 ⊢ 1 ≤ ↑↑μH[0] s ** exact one_le_hausdorffMeasure_zero_of_nonempty ⟨x, hx⟩ ** case inl.inr.intro.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r d : ℝ hd : 0 ≤ d h : HolderOnWith 0 r f s x : X hx : x ∈ s this : f '' s = {f x} h'd : 0 < d ⊢ ↑↑μH[d] {f x} ≤ ↑0 ^ d * ↑↑μH[↑r * d] s ** haveI := noAtoms_hausdorff Y h'd ** case inl.inr.intro.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y r : ℝ≥0 f : X → Y s t : Set X hr : 0 < r d : ℝ hd : 0 ≤ d h : HolderOnWith 0 r f s x : X hx : x ∈ s this✝ : f '' s = {f x} h'd : 0 < d this : NoAtoms μH[d] ⊢ ↑↑μH[d] {f x} ≤ ↑0 ^ d * ↑↑μH[↑r * d] s ** simp only [zero_le, measure_singleton] ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C ⊢ ↑↑μH[d] (f '' s) ≤ ↑C ^ d * ↑↑μH[↑r * d] s ** have hCd0 : (C : ℝ≥0∞) ^ d ≠ 0 := by simp [hC0.ne'] ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 ⊢ ↑↑μH[d] (f '' s) ≤ ↑C ^ d * ↑↑μH[↑r * d] s ** have hCd : (C : ℝ≥0∞) ^ d ≠ ∞ := by simp [hd] ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ ⊢ ↑↑μH[d] (f '' s) ≤ ↑C ^ d * ↑↑μH[↑r * d] s ** simp only [hausdorffMeasure_apply, ENNReal.mul_iSup, ENNReal.mul_iInf_of_ne hCd0 hCd,
← ENNReal.tsum_mul_left] ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ ⊢ ⨆ r, ⨆ (_ : 0 < r), ⨅ t, ⨅ (_ : f '' s ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ d ≤ ⨆ i, ⨆ (_ : 0 < i), ⨅ i_1, ⨅ (_ : s ⊆ ⋃ n, i_1 n), ⨅ (_ : ∀ (n : ℕ), diam (i_1 n) ≤ i), ∑' (i : ℕ), ⨆ (_ : Set.Nonempty (i_1 i)), ↑C ^ d * diam (i_1 i) ^ (↑r * d) ** refine' iSup_le fun R => iSup_le fun hR => _ ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R ⊢ ⨅ t, ⨅ (_ : f '' s ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ R), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ d ≤ ⨆ i, ⨆ (_ : 0 < i), ⨅ i_1, ⨅ (_ : s ⊆ ⋃ n, i_1 n), ⨅ (_ : ∀ (n : ℕ), diam (i_1 n) ≤ i), ∑' (i : ℕ), ⨆ (_ : Set.Nonempty (i_1 i)), ↑C ^ d * diam (i_1 i) ^ (↑r * d) ** have : Tendsto (fun d : ℝ≥0∞ => (C : ℝ≥0∞) * d ^ (r : ℝ)) (𝓝 0) (𝓝 0) :=
ENNReal.tendsto_const_mul_rpow_nhds_zero_of_pos ENNReal.coe_ne_top hr ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) ⊢ ⨅ t, ⨅ (_ : f '' s ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ R), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ d ≤ ⨆ i, ⨆ (_ : 0 < i), ⨅ i_1, ⨅ (_ : s ⊆ ⋃ n, i_1 n), ⨅ (_ : ∀ (n : ℕ), diam (i_1 n) ≤ i), ∑' (i : ℕ), ⨆ (_ : Set.Nonempty (i_1 i)), ↑C ^ d * diam (i_1 i) ^ (↑r * d) ** rcases ENNReal.nhds_zero_basis_Iic.eventually_iff.1 (this.eventually (gt_mem_nhds hR)) with
⟨δ, δ0, H⟩ ** case inr.intro.intro ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) δ : ℝ≥0∞ δ0 : 0 < δ H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R ⊢ ⨅ t, ⨅ (_ : f '' s ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ R), ∑' (n : ℕ), ⨆ (_ : Set.Nonempty (t n)), diam (t n) ^ d ≤ ⨆ i, ⨆ (_ : 0 < i), ⨅ i_1, ⨅ (_ : s ⊆ ⋃ n, i_1 n), ⨅ (_ : ∀ (n : ℕ), diam (i_1 n) ≤ i), ∑' (i : ℕ), ⨆ (_ : Set.Nonempty (i_1 i)), ↑C ^ d * diam (i_1 i) ^ (↑r * d) ** refine'
le_iSup₂_of_le δ δ0
(iInf₂_mono' fun t hst =>
⟨fun n => f '' (t n ∩ s), _,
iInf_mono' fun htδ =>
⟨fun n => (h.ediam_image_inter_le (t n)).trans (H (htδ n)).le, _⟩⟩) ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C ⊢ ↑C ^ d ≠ 0 ** simp [hC0.ne'] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 ⊢ ↑C ^ d ≠ ⊤ ** simp [hd] ** case inr.intro.intro.refine'_1 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t✝ : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) δ : ℝ≥0∞ δ0 : 0 < δ H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R t : ℕ → Set X hst : s ⊆ ⋃ n, t n ⊢ f '' s ⊆ ⋃ n, (fun n => f '' (t n ∩ s)) n ** rw [← image_iUnion, ← iUnion_inter] ** case inr.intro.intro.refine'_1 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t✝ : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) δ : ℝ≥0∞ δ0 : 0 < δ H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R t : ℕ → Set X hst : s ⊆ ⋃ n, t n ⊢ f '' s ⊆ f '' ((⋃ i, t i) ∩ s) ** exact image_subset _ (subset_inter hst Subset.rfl) ** case inr.intro.intro.refine'_2 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t✝ : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) δ : ℝ≥0∞ δ0 : 0 < δ H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R t : ℕ → Set X hst : s ⊆ ⋃ n, t n htδ : ∀ (n : ℕ), diam (t n) ≤ δ ⊢ ∑' (n : ℕ), ⨆ (_ : Set.Nonempty ((fun n => f '' (t n ∩ s)) n)), diam ((fun n => f '' (t n ∩ s)) n) ^ d ≤ ∑' (i : ℕ), ⨆ (_ : Set.Nonempty (t i)), ↑C ^ d * diam (t i) ^ (↑r * d) ** refine' ENNReal.tsum_le_tsum fun n => _ ** case inr.intro.intro.refine'_2 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t✝ : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) δ : ℝ≥0∞ δ0 : 0 < δ H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R t : ℕ → Set X hst : s ⊆ ⋃ n, t n htδ : ∀ (n : ℕ), diam (t n) ≤ δ n : ℕ ⊢ ⨆ (_ : Set.Nonempty ((fun n => f '' (t n ∩ s)) n)), diam ((fun n => f '' (t n ∩ s)) n) ^ d ≤ ⨆ (_ : Set.Nonempty (t n)), ↑C ^ d * diam (t n) ^ (↑r * d) ** simp only [iSup_le_iff, nonempty_image_iff] ** case inr.intro.intro.refine'_2 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t✝ : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) δ : ℝ≥0∞ δ0 : 0 < δ H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R t : ℕ → Set X hst : s ⊆ ⋃ n, t n htδ : ∀ (n : ℕ), diam (t n) ≤ δ n : ℕ ⊢ Set.Nonempty (t n ∩ s) → diam (f '' (t n ∩ s)) ^ d ≤ ⨆ (_ : Set.Nonempty (t n)), ↑C ^ d * diam (t n) ^ (↑r * d) ** intro hft ** case inr.intro.intro.refine'_2 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t✝ : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) δ : ℝ≥0∞ δ0 : 0 < δ H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R t : ℕ → Set X hst : s ⊆ ⋃ n, t n htδ : ∀ (n : ℕ), diam (t n) ≤ δ n : ℕ hft : Set.Nonempty (t n ∩ s) ⊢ diam (f '' (t n ∩ s)) ^ d ≤ ⨆ (_ : Set.Nonempty (t n)), ↑C ^ d * diam (t n) ^ (↑r * d) ** simp only [Nonempty.mono ((t n).inter_subset_left s) hft, ciSup_pos] ** case inr.intro.intro.refine'_2 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t✝ : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) δ : ℝ≥0∞ δ0 : 0 < δ H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R t : ℕ → Set X hst : s ⊆ ⋃ n, t n htδ : ∀ (n : ℕ), diam (t n) ≤ δ n : ℕ hft : Set.Nonempty (t n ∩ s) ⊢ diam (f '' (t n ∩ s)) ^ d ≤ ↑C ^ d * diam (t n) ^ (↑r * d) ** rw [ENNReal.rpow_mul, ← ENNReal.mul_rpow_of_nonneg _ _ hd] ** case inr.intro.intro.refine'_2 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y C r : ℝ≥0 f : X → Y s t✝ : Set X h : HolderOnWith C r f s hr : 0 < r d : ℝ hd : 0 ≤ d hC0 : 0 < C hCd0 : ↑C ^ d ≠ 0 hCd : ↑C ^ d ≠ ⊤ R : ℝ≥0∞ hR : 0 < R this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0) δ : ℝ≥0∞ δ0 : 0 < δ H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R t : ℕ → Set X hst : s ⊆ ⋃ n, t n htδ : ∀ (n : ℕ), diam (t n) ≤ δ n : ℕ hft : Set.Nonempty (t n ∩ s) ⊢ diam (f '' (t n ∩ s)) ^ d ≤ (↑C * diam (t n) ^ ↑r) ^ d ** exact ENNReal.rpow_le_rpow (h.ediam_image_inter_le _) hd ** Qed
| |
LipschitzOnWith.hausdorffMeasure_image_le ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y K : ℝ≥0 f : X → Y s t : Set X h : LipschitzOnWith K f s d : ℝ hd : 0 ≤ d ⊢ ↑↑μH[d] (f '' s) ≤ ↑K ^ d * ↑↑μH[d] s ** simpa only [NNReal.coe_one, one_mul] using h.holderOnWith.hausdorffMeasure_image_le zero_lt_one hd ** Qed
| |
Isometry.hausdorffMeasure_preimage ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y f : X → Y d : ℝ hf : Isometry f hd : 0 ≤ d ∨ Surjective f s : Set Y ⊢ ↑↑μH[d] (f ⁻¹' s) = ↑↑μH[d] (s ∩ range f) ** rw [← hf.hausdorffMeasure_image hd, image_preimage_eq_inter_range] ** Qed
| |
Isometry.map_hausdorffMeasure ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y f : X → Y d : ℝ hf : Isometry f hd : 0 ≤ d ∨ Surjective f ⊢ Measure.map f μH[d] = Measure.restrict μH[d] (range f) ** ext1 s hs ** case h ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y f : X → Y d : ℝ hf : Isometry f hd : 0 ≤ d ∨ Surjective f s : Set Y hs : MeasurableSet s ⊢ ↑↑(Measure.map f μH[d]) s = ↑↑(Measure.restrict μH[d] (range f)) s ** rw [map_apply hf.continuous.measurable hs, Measure.restrict_apply hs,
hf.hausdorffMeasure_preimage hd] ** Qed
| |
IsometryEquiv.hausdorffMeasure_preimage ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y e : X ≃ᵢ Y d : ℝ s : Set Y ⊢ ↑↑μH[d] (↑e ⁻¹' s) = ↑↑μH[d] s ** rw [← e.image_symm, e.symm.hausdorffMeasure_image] ** Qed
| |
IsometryEquiv.map_hausdorffMeasure ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁵ : EMetricSpace X inst✝⁴ : EMetricSpace Y inst✝³ : MeasurableSpace X inst✝² : BorelSpace X inst✝¹ : MeasurableSpace Y inst✝ : BorelSpace Y e : X ≃ᵢ Y d : ℝ ⊢ Measure.map ↑e μH[d] = μH[d] ** rw [e.isometry.map_hausdorffMeasure (Or.inr e.surjective), e.surjective.range_eq, restrict_univ] ** Qed
| |
MeasureTheory.hausdorffMeasure_smul_right_image ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ ⊢ ↑↑μH[1] ((fun r => r • v) '' s) = ‖v‖₊ • ↑↑μH[1] s ** obtain rfl | hv := eq_or_ne v 0 ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 ⊢ ↑↑μH[1] ((fun r => r • v) '' s) = ‖v‖₊ • ↑↑μH[1] s ** have hn : ‖v‖ ≠ 0 := norm_ne_zero_iff.mpr hv ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 ⊢ ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s ** have iso_smul : Isometry (LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) := by
refine' AddMonoidHomClass.isometry_of_norm _ fun x => (norm_smul _ _).trans _
rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel hn, mul_one, LinearMap.id_apply] ** case inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 iso_smul : Isometry ↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) ⊢ ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s ** rw [Set.image_smul, Measure.hausdorffMeasure_smul₀ zero_le_one hn, nnnorm_norm,
NNReal.rpow_eq_pow, NNReal.rpow_one,
iso_smul.hausdorffMeasure_image (Or.inl <| zero_le_one' ℝ)] ** case inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E s : Set ℝ ⊢ ↑↑μH[1] ((fun r => r • 0) '' s) = ‖0‖₊ • ↑↑μH[1] s ** haveI := noAtoms_hausdorff E one_pos ** case inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E s : Set ℝ this : NoAtoms μH[1] ⊢ ↑↑μH[1] ((fun r => r • 0) '' s) = ‖0‖₊ • ↑↑μH[1] s ** obtain rfl | hs := s.eq_empty_or_nonempty ** case inl.inr ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E s : Set ℝ this : NoAtoms μH[1] hs : Set.Nonempty s ⊢ ↑↑μH[1] ((fun r => r • 0) '' s) = ‖0‖₊ • ↑↑μH[1] s ** simp [hs] ** case inl.inl ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E this : NoAtoms μH[1] ⊢ ↑↑μH[1] ((fun r => r • 0) '' ∅) = ‖0‖₊ • ↑↑μH[1] ∅ ** simp ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s ⊢ ↑↑μH[1] ((fun r => r • v) '' s) = ‖v‖₊ • ↑↑μH[1] s ** simp only [hausdorffMeasure_real, nnreal_smul_coe_apply] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s ⊢ ↑↑μH[1] ((fun r => r • v) '' s) = ↑‖v‖₊ * ↑↑volume s ** convert this ** case h.e'_2.h.e'_3 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s ⊢ (fun r => r • v) '' s = (fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s) ** simp only [image_smul, LinearMap.toSpanSingleton_apply, Set.image_image] ** case h.e'_2.h.e'_3 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s ⊢ (fun r => r • v) '' s = (fun a => ‖v‖ • a • ‖v‖⁻¹ • v) '' s ** ext e ** case h.e'_2.h.e'_3.h ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s e : E ⊢ e ∈ (fun r => r • v) '' s ↔ e ∈ (fun a => ‖v‖ • a • ‖v‖⁻¹ • v) '' s ** simp only [mem_image] ** case h.e'_2.h.e'_3.h ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s e : E ⊢ (∃ x, x ∈ s ∧ x • v = e) ↔ ∃ x, x ∈ s ∧ ‖v‖ • x • ‖v‖⁻¹ • v = e ** refine' ⟨fun ⟨x, h⟩ => ⟨x, _⟩, fun ⟨x, h⟩ => ⟨x, _⟩⟩ ** case h.e'_2.h.e'_3.h.refine'_1 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s e : E x✝ : ∃ x, x ∈ s ∧ x • v = e x : ℝ h : x ∈ s ∧ x • v = e ⊢ x ∈ s ∧ ‖v‖ • x • ‖v‖⁻¹ • v = e ** rw [smul_comm (norm _), smul_comm (norm _), inv_smul_smul₀ hn] ** case h.e'_2.h.e'_3.h.refine'_1 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s e : E x✝ : ∃ x, x ∈ s ∧ x • v = e x : ℝ h : x ∈ s ∧ x • v = e ⊢ x ∈ s ∧ x • v = e ** exact h ** case h.e'_2.h.e'_3.h.refine'_2 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s e : E x✝ : ∃ x, x ∈ s ∧ ‖v‖ • x • ‖v‖⁻¹ • v = e x : ℝ h : x ∈ s ∧ ‖v‖ • x • ‖v‖⁻¹ • v = e ⊢ x ∈ s ∧ x • v = e ** rw [smul_comm (norm _), smul_comm (norm _), inv_smul_smul₀ hn] at h ** case h.e'_2.h.e'_3.h.refine'_2 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s e : E x✝ : ∃ x, x ∈ s ∧ ‖v‖ • x • ‖v‖⁻¹ • v = e x : ℝ h : x ∈ s ∧ x • v = e ⊢ x ∈ s ∧ x • v = e ** exact h ** case h.e'_3.h.e'_1.h.e'_2.h.e'_3 ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 this : ↑↑μH[1] ((fun x x_1 => x • x_1) ‖v‖ '' (↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s)) = ‖v‖₊ • ↑↑μH[1] s ⊢ volume = μH[1] ** exact hausdorffMeasure_real.symm ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 ⊢ Isometry ↑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) ** refine' AddMonoidHomClass.isometry_of_norm _ fun x => (norm_smul _ _).trans _ ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝⁹ : EMetricSpace X inst✝⁸ : EMetricSpace Y inst✝⁷ : MeasurableSpace X inst✝⁶ : BorelSpace X inst✝⁵ : MeasurableSpace Y inst✝⁴ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E inst✝¹ : MeasurableSpace E inst✝ : BorelSpace E v : E s : Set ℝ hv : v ≠ 0 hn : ‖v‖ ≠ 0 x : ℝ ⊢ ‖↑LinearMap.id x‖ * ‖‖v‖⁻¹ • v‖ = ‖x‖ ** rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel hn, mul_one, LinearMap.id_apply] ** Qed
| |
MeasureTheory.hausdorffMeasure_homothety_image ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹² : EMetricSpace X inst✝¹¹ : EMetricSpace Y inst✝¹⁰ : MeasurableSpace X inst✝⁹ : BorelSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝⁶ : NormedField 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : MeasurableSpace P inst✝² : MetricSpace P inst✝¹ : NormedAddTorsor E P inst✝ : BorelSpace P d : ℝ hd : 0 ≤ d x : P c : 𝕜 hc : c ≠ 0 s : Set P ⊢ ↑↑μH[d] (↑(IsometryEquiv.vaddConst x) '' ((fun x x_1 => x • x_1) c '' (↑(IsometryEquiv.symm (IsometryEquiv.vaddConst x)) '' s))) = NNReal.rpow ‖c‖₊ d • ↑↑μH[d] s ** borelize E ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹² : EMetricSpace X inst✝¹¹ : EMetricSpace Y inst✝¹⁰ : MeasurableSpace X inst✝⁹ : BorelSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝⁶ : NormedField 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : MeasurableSpace P inst✝² : MetricSpace P inst✝¹ : NormedAddTorsor E P inst✝ : BorelSpace P d : ℝ hd : 0 ≤ d x : P c : 𝕜 hc : c ≠ 0 s : Set P this✝¹ : MeasurableSpace E := borel E this✝ : BorelSpace E ⊢ ↑↑μH[d] (↑(IsometryEquiv.vaddConst x) '' ((fun x x_1 => x • x_1) c '' (↑(IsometryEquiv.symm (IsometryEquiv.vaddConst x)) '' s))) = NNReal.rpow ‖c‖₊ d • ↑↑μH[d] s ** rw [IsometryEquiv.hausdorffMeasure_image, Set.image_smul, Measure.hausdorffMeasure_smul₀ hd hc,
IsometryEquiv.hausdorffMeasure_image] ** ι : Type u_1 X : Type u_2 Y : Type u_3 inst✝¹² : EMetricSpace X inst✝¹¹ : EMetricSpace Y inst✝¹⁰ : MeasurableSpace X inst✝⁹ : BorelSpace X inst✝⁸ : MeasurableSpace Y inst✝⁷ : BorelSpace Y 𝕜 : Type u_4 E : Type u_5 P : Type u_6 inst✝⁶ : NormedField 𝕜 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace 𝕜 E inst✝³ : MeasurableSpace P inst✝² : MetricSpace P inst✝¹ : NormedAddTorsor E P inst✝ : BorelSpace P d : ℝ hd : 0 ≤ d x : P c : 𝕜 hc : c ≠ 0 s : Set P this : ↑↑μH[d] (↑(IsometryEquiv.vaddConst x) '' ((fun x x_1 => x • x_1) c '' (↑(IsometryEquiv.symm (IsometryEquiv.vaddConst x)) '' s))) = NNReal.rpow ‖c‖₊ d • ↑↑μH[d] s ⊢ ↑↑μH[d] (↑(AffineMap.homothety x c) '' s) = NNReal.rpow ‖c‖₊ d • ↑↑μH[d] s ** simpa only [Set.image_image] ** Qed
| |
MeasureTheory.Measure.MutuallySingular.mk ** α : Type u_1 m0 : MeasurableSpace α μ μ₁ μ₂ ν ν₁ ν₂ : Measure α s t : Set α hs : ↑↑μ s = 0 ht : ↑↑ν t = 0 hst : univ ⊆ s ∪ t ⊢ μ ⟂ₘ ν ** use toMeasurable μ s, measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans hs ** case right α : Type u_1 m0 : MeasurableSpace α μ μ₁ μ₂ ν ν₁ ν₂ : Measure α s t : Set α hs : ↑↑μ s = 0 ht : ↑↑ν t = 0 hst : univ ⊆ s ∪ t ⊢ ↑↑ν (toMeasurable μ s)ᶜ = 0 ** refine' measure_mono_null (fun x hx => (hst trivial).resolve_left fun hxs => hx _) ht ** case right α : Type u_1 m0 : MeasurableSpace α μ μ₁ μ₂ ν ν₁ ν₂ : Measure α s t : Set α hs : ↑↑μ s = 0 ht : ↑↑ν t = 0 hst : univ ⊆ s ∪ t x : α hx : x ∈ (toMeasurable μ s)ᶜ hxs : x ∈ s ⊢ x ∈ toMeasurable μ s ** exact subset_toMeasurable _ _ hxs ** Qed
| |
MeasureTheory.Measure.MutuallySingular.sum_left ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α ι : Type u_2 inst✝ : Countable ι μ : ι → Measure α ⊢ sum μ ⟂ₘ ν ↔ ∀ (i : ι), μ i ⟂ₘ ν ** refine' ⟨fun h i => h.mono (le_sum _ _) le_rfl, fun H => _⟩ ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α ι : Type u_2 inst✝ : Countable ι μ : ι → Measure α H : ∀ (i : ι), μ i ⟂ₘ ν ⊢ sum μ ⟂ₘ ν ** choose s hsm hsμ hsν using H ** α : Type u_1 m0 : MeasurableSpace α μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α ι : Type u_2 inst✝ : Countable ι μ : ι → Measure α s : ι → Set α hsm : ∀ (i : ι), MeasurableSet (s i) hsμ : ∀ (i : ι), ↑↑(μ i) (s i) = 0 hsν : ∀ (i : ι), ↑↑ν (s i)ᶜ = 0 ⊢ sum μ ⟂ₘ ν ** refine' ⟨⋂ i, s i, MeasurableSet.iInter hsm, _, _⟩ ** case refine'_1 α : Type u_1 m0 : MeasurableSpace α μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α ι : Type u_2 inst✝ : Countable ι μ : ι → Measure α s : ι → Set α hsm : ∀ (i : ι), MeasurableSet (s i) hsμ : ∀ (i : ι), ↑↑(μ i) (s i) = 0 hsν : ∀ (i : ι), ↑↑ν (s i)ᶜ = 0 ⊢ ↑↑(sum μ) (⋂ i, s i) = 0 ** rw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero] ** case refine'_1 α : Type u_1 m0 : MeasurableSpace α μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α ι : Type u_2 inst✝ : Countable ι μ : ι → Measure α s : ι → Set α hsm : ∀ (i : ι), MeasurableSet (s i) hsμ : ∀ (i : ι), ↑↑(μ i) (s i) = 0 hsν : ∀ (i : ι), ↑↑ν (s i)ᶜ = 0 ⊢ ∀ (i : ι), ↑↑(μ i) (⋂ b, s b) = 0 ** exact fun i => measure_mono_null (iInter_subset _ _) (hsμ i) ** case refine'_2 α : Type u_1 m0 : MeasurableSpace α μ✝ μ₁ μ₂ ν ν₁ ν₂ : Measure α ι : Type u_2 inst✝ : Countable ι μ : ι → Measure α s : ι → Set α hsm : ∀ (i : ι), MeasurableSet (s i) hsμ : ∀ (i : ι), ↑↑(μ i) (s i) = 0 hsν : ∀ (i : ι), ↑↑ν (s i)ᶜ = 0 ⊢ ↑↑ν (⋂ i, s i)ᶜ = 0 ** rwa [compl_iInter, measure_iUnion_null_iff] ** Qed
| |
MeasureTheory.withDensityᵥ_apply ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f s : Set α hs : MeasurableSet s ⊢ ↑(withDensityᵥ μ f) s = ∫ (x : α) in s, f x ∂μ ** rw [withDensityᵥ, dif_pos hf] ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E hf : Integrable f s : Set α hs : MeasurableSet s ⊢ ↑{ measureOf' := fun s => if MeasurableSet s then ∫ (x : α) in s, f x ∂μ else 0, empty' := (_ : (if MeasurableSet ∅ then ∫ (x : α) in ∅, f x ∂μ else 0) = 0), not_measurable' := (_ : ∀ (s : Set α), ¬MeasurableSet s → (if MeasurableSet s then ∫ (x : α) in s, f x ∂μ else 0) = 0), m_iUnion' := (_ : ∀ (s : ℕ → Set α), (∀ (i : ℕ), MeasurableSet (s i)) → Pairwise (Disjoint on s) → HasSum (fun i => (fun s => if MeasurableSet s then ∫ (x : α) in s, f x ∂μ else 0) (s i)) ((fun s => if MeasurableSet s then ∫ (x : α) in s, f x ∂μ else 0) (⋃ i, s i))) } s = ∫ (x : α) in s, f x ∂μ ** exact dif_pos hs ** Qed
| |
MeasureTheory.withDensityᵥ_zero ** α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E ⊢ withDensityᵥ μ 0 = 0 ** ext1 s hs ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E s : Set α hs : MeasurableSet s ⊢ ↑(withDensityᵥ μ 0) s = ↑0 s ** erw [withDensityᵥ_apply (integrable_zero α E μ) hs] ** case h α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α E : Type u_3 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f g : α → E s : Set α hs : MeasurableSet s ⊢ ∫ (x : α) in s, 0 ∂μ = ↑0 s ** simp ** Qed
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.