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interval_average_symm ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f : ℝ → E a b : ℝ ⊢ ⨍ (x : ℝ) in a..b, f x = ⨍ (x : ℝ) in b..a, f x ** rw [setAverage_eq, setAverage_eq, uIoc_comm] ** Qed
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interval_average_eq ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f : ℝ → E a b : ℝ ⊢ ⨍ (x : ℝ) in a..b, f x = (b - a)⁻¹ • ∫ (x : ℝ) in a..b, f x ** cases' le_or_lt a b with h h ** case inl E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f : ℝ → E a b : ℝ h : a ≤ b ⊢ ⨍ (x : ℝ) in a..b, f x = (b - a)⁻¹ • ∫ (x : ℝ) in a..b, f x ** rw [setAverage_eq, uIoc_of_le h, Real.volume_Ioc, intervalIntegral.integral_of_le h,
ENNReal.toReal_ofReal (sub_nonneg.2 h)] ** case inr E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f : ℝ → E a b : ℝ h : b < a ⊢ ⨍ (x : ℝ) in a..b, f x = (b - a)⁻¹ • ∫ (x : ℝ) in a..b, f x ** rw [setAverage_eq, uIoc_of_lt h, Real.volume_Ioc, intervalIntegral.integral_of_ge h.le,
ENNReal.toReal_ofReal (sub_nonneg.2 h.le), smul_neg, ← neg_smul, ← inv_neg, neg_sub] ** Qed
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periodic_circleMap ** E : Type u_1 inst✝ : NormedAddCommGroup E c : ℂ R θ : ℝ ⊢ circleMap c R (θ + 2 * π) = circleMap c R θ ** simp [circleMap, add_mul, exp_periodic _] ** Qed
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circleMap_sub_center ** E : Type u_1 inst✝ : NormedAddCommGroup E c : ℂ R θ : ℝ ⊢ circleMap c R θ - c = circleMap 0 R θ ** simp [circleMap] ** Qed
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abs_circleMap_zero ** E : Type u_1 inst✝ : NormedAddCommGroup E R θ : ℝ ⊢ ↑Complex.abs (circleMap 0 R θ) = |R| ** simp [circleMap] ** Qed
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circleMap_mem_sphere' ** E : Type u_1 inst✝ : NormedAddCommGroup E c : ℂ R θ : ℝ ⊢ circleMap c R θ ∈ sphere c |R| ** simp ** Qed
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circleMap_mem_sphere ** E : Type u_1 inst✝ : NormedAddCommGroup E c : ℂ R : ℝ hR : 0 ≤ R θ : ℝ ⊢ circleMap c R θ ∈ sphere c R ** simpa only [_root_.abs_of_nonneg hR] using circleMap_mem_sphere' c R θ ** Qed
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circleMap_not_mem_ball ** E : Type u_1 inst✝ : NormedAddCommGroup E c : ℂ R θ : ℝ ⊢ ¬circleMap c R θ ∈ ball c R ** simp [dist_eq, le_abs_self] ** Qed
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circleMap_eq_center_iff ** E : Type u_1 inst✝ : NormedAddCommGroup E c : ℂ R θ : ℝ ⊢ circleMap c R θ = c ↔ R = 0 ** simp [circleMap, exp_ne_zero] ** Qed
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hasDerivAt_circleMap ** E : Type u_1 inst✝ : NormedAddCommGroup E c : ℂ R θ : ℝ ⊢ HasDerivAt (circleMap c R) (circleMap 0 R θ * I) θ ** simpa only [mul_assoc, one_mul, ofRealClm_apply, circleMap, ofReal_one, zero_add]
using (((ofRealClm.hasDerivAt (x := θ)).mul_const I).cexp.const_mul (R : ℂ)).const_add c ** Qed
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deriv_circleMap_eq_zero_iff ** E : Type u_1 inst✝ : NormedAddCommGroup E c : ℂ R θ : ℝ ⊢ deriv (circleMap c R) θ = 0 ↔ R = 0 ** simp [I_ne_zero] ** Qed
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continuous_circleMap_inv ** E : Type u_1 inst✝ : NormedAddCommGroup E R : ℝ z w : ℂ hw : w ∈ ball z R ⊢ Continuous fun θ => (circleMap z R θ - w)⁻¹ ** have : ∀ θ, circleMap z R θ - w ≠ 0 := by
simp_rw [sub_ne_zero]
exact fun θ => circleMap_ne_mem_ball hw θ ** E : Type u_1 inst✝ : NormedAddCommGroup E R : ℝ z w : ℂ hw : w ∈ ball z R this : ∀ (θ : ℝ), circleMap z R θ - w ≠ 0 ⊢ Continuous fun θ => (circleMap z R θ - w)⁻¹ ** exact Continuous.inv₀ (by continuity) this ** E : Type u_1 inst✝ : NormedAddCommGroup E R : ℝ z w : ℂ hw : w ∈ ball z R ⊢ ∀ (θ : ℝ), circleMap z R θ - w ≠ 0 ** simp_rw [sub_ne_zero] ** E : Type u_1 inst✝ : NormedAddCommGroup E R : ℝ z w : ℂ hw : w ∈ ball z R ⊢ ∀ (θ : ℝ), circleMap z R θ ≠ w ** exact fun θ => circleMap_ne_mem_ball hw θ ** E : Type u_1 inst✝ : NormedAddCommGroup E R : ℝ z w : ℂ hw : w ∈ ball z R this : ∀ (θ : ℝ), circleMap z R θ - w ≠ 0 ⊢ Continuous fun θ => circleMap z R θ - w ** continuity ** Qed
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CircleIntegrable.out ** E : Type u_1 inst✝¹ : NormedAddCommGroup E f g : ℂ → E c : ℂ R : ℝ inst✝ : NormedSpace ℂ E hf : CircleIntegrable f c R ⊢ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π) ** simp only [CircleIntegrable, deriv_circleMap, intervalIntegrable_iff] at * ** E : Type u_1 inst✝¹ : NormedAddCommGroup E f g : ℂ → E c : ℂ R : ℝ inst✝ : NormedSpace ℂ E hf : IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) ⊢ IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) ** refine' (hf.norm.const_mul |R|).mono' _ _ ** case refine'_1 E : Type u_1 inst✝¹ : NormedAddCommGroup E f g : ℂ → E c : ℂ R : ℝ inst✝ : NormedSpace ℂ E hf : IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) ⊢ AEStronglyMeasurable (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Measure.restrict volume (Ι 0 (2 * π))) ** exact ((continuous_circleMap _ _).aestronglyMeasurable.mul_const I).smul hf.aestronglyMeasurable ** case refine'_2 E : Type u_1 inst✝¹ : NormedAddCommGroup E f g : ℂ → E c : ℂ R : ℝ inst✝ : NormedSpace ℂ E hf : IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) ⊢ ∀ᵐ (a : ℝ) ∂Measure.restrict volume (Ι 0 (2 * π)), ‖(circleMap 0 R a * I) • f (circleMap c R a)‖ ≤ |R| * ‖f (circleMap c R a)‖ ** simp [norm_smul] ** Qed
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circleIntegrable_zero_radius ** E : Type u_1 inst✝ : NormedAddCommGroup E f : ℂ → E c : ℂ ⊢ CircleIntegrable f c 0 ** simp [CircleIntegrable] ** Qed
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circleIntegrable_iff ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E f : ℂ → E c : ℂ R : ℝ ⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π) ** by_cases h₀ : R = 0 ** case neg E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E f : ℂ → E c : ℂ R : ℝ h₀ : ¬R = 0 ⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π) ** refine' ⟨fun h => h.out, fun h => _⟩ ** case neg E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E f : ℂ → E c : ℂ R : ℝ h₀ : ¬R = 0 h : IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π) ⊢ CircleIntegrable f c R ** simp only [CircleIntegrable, intervalIntegrable_iff, deriv_circleMap] at h ⊢ ** case neg E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E f : ℂ → E c : ℂ R : ℝ h₀ : ¬R = 0 h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) ⊢ IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) ** refine' (h.norm.const_mul |R|⁻¹).mono' _ _ ** case pos E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E f : ℂ → E c : ℂ R : ℝ h₀ : R = 0 ⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π) ** simp [h₀, const] ** case neg.refine'_1 E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E f : ℂ → E c : ℂ R : ℝ h₀ : ¬R = 0 h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) ⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (Measure.restrict volume (Ι 0 (2 * π))) ** have H : ∀ {θ}, circleMap 0 R θ * I ≠ 0 := fun {θ} => by simp [h₀, I_ne_zero] ** case neg.refine'_1 E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E f : ℂ → E c : ℂ R : ℝ h₀ : ¬R = 0 h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) H : ∀ {θ : ℝ}, circleMap 0 R θ * I ≠ 0 ⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (Measure.restrict volume (Ι 0 (2 * π))) ** simpa only [inv_smul_smul₀ H]
using ((continuous_circleMap 0 R).aestronglyMeasurable.mul_const
I).aemeasurable.inv.aestronglyMeasurable.smul h.aestronglyMeasurable ** E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E f : ℂ → E c : ℂ R : ℝ h₀ : ¬R = 0 h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) θ : ℝ ⊢ circleMap 0 R θ * I ≠ 0 ** simp [h₀, I_ne_zero] ** case neg.refine'_2 E : Type u_1 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℂ E f : ℂ → E c : ℂ R : ℝ h₀ : ¬R = 0 h : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) ⊢ ∀ᵐ (a : ℝ) ∂Measure.restrict volume (Ι 0 (2 * π)), ‖f (circleMap c R a)‖ ≤ |R|⁻¹ * ‖(circleMap 0 R a * I) • f (circleMap c R a)‖ ** simp [norm_smul, h₀] ** Qed
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circleIntegrable_sub_inv_iff ** E : Type u_1 inst✝ : NormedAddCommGroup E c w : ℂ R : ℝ ⊢ CircleIntegrable (fun z => (z - w)⁻¹) c R ↔ R = 0 ∨ ¬w ∈ sphere c |R| ** simp only [← zpow_neg_one, circleIntegrable_sub_zpow_iff] ** E : Type u_1 inst✝ : NormedAddCommGroup E c w : ℂ R : ℝ ⊢ R = 0 ∨ False ∨ ¬w ∈ sphere c |R| ↔ R = 0 ∨ ¬w ∈ sphere c |R| ** norm_num ** Qed
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circleIntegral_def_Icc ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ ⊢ (∮ (z : ℂ) in C(c, R), f z) = ∫ (θ : ℝ) in Icc 0 (2 * π), deriv (circleMap c R) θ • f (circleMap c R θ) ** rw [circleIntegral, intervalIntegral.integral_of_le Real.two_pi_pos.le,
Measure.restrict_congr_set Ioc_ae_eq_Icc] ** Qed
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circleIntegral.integral_radius_zero ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ ⊢ (∮ (z : ℂ) in C(c, 0), f z) = 0 ** simp [circleIntegral, const] ** Qed
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circleIntegral.integral_congr ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f g : ℂ → E c : ℂ R : ℝ hR : 0 ≤ R h : EqOn f g (sphere c R) θ : ℝ x✝ : θ ∈ [[0, 2 * π]] ⊢ deriv (circleMap c R) θ • (fun z => f z) (circleMap c R θ) = deriv (circleMap c R) θ • (fun z => g z) (circleMap c R θ) ** simp only [h (circleMap_mem_sphere _ hR _)] ** Qed
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circleIntegral.integral_sub ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f g : ℂ → E c : ℂ R : ℝ hf : CircleIntegrable f c R hg : CircleIntegrable g c R ⊢ (∮ (z : ℂ) in C(c, R), f z - g z) = (∮ (z : ℂ) in C(c, R), f z) - ∮ (z : ℂ) in C(c, R), g z ** simp only [circleIntegral, smul_sub, intervalIntegral.integral_sub hf.out hg.out] ** Qed
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circleIntegral.norm_integral_le_of_norm_le_const' ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R C : ℝ hf : ∀ (z : ℂ), z ∈ sphere c |R| → ‖f z‖ ≤ C θ : ℝ x✝ : θ ∈ Ι 0 (2 * π) ⊢ ‖deriv (circleMap c R) θ • f (circleMap c R θ)‖ = |R| * ‖f (circleMap c R θ)‖ ** simp [norm_smul] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R C : ℝ hf : ∀ (z : ℂ), z ∈ sphere c |R| → ‖f z‖ ≤ C ⊢ |R| * C * |2 * π - 0| = 2 * π * |R| * C ** rw [sub_zero, _root_.abs_of_pos Real.two_pi_pos] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R C : ℝ hf : ∀ (z : ℂ), z ∈ sphere c |R| → ‖f z‖ ≤ C ⊢ |R| * C * (2 * π) = 2 * π * |R| * C ** ac_rfl ** Qed
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circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R C : ℝ hR : 0 ≤ R hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C ⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), f z‖ ≤ R * C ** have : ‖(2 * π * I : ℂ)⁻¹‖ = (2 * π)⁻¹ := by simp [Real.pi_pos.le] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R C : ℝ hR : 0 ≤ R hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C this : ‖(2 * ↑π * I)⁻¹‖ = (2 * π)⁻¹ ⊢ ‖(2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), f z‖ ≤ R * C ** rw [norm_smul, this, ← div_eq_inv_mul, div_le_iff Real.two_pi_pos, mul_comm (R * C), ← mul_assoc] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R C : ℝ hR : 0 ≤ R hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C this : ‖(2 * ↑π * I)⁻¹‖ = (2 * π)⁻¹ ⊢ ‖∮ (z : ℂ) in C(c, R), f z‖ ≤ 2 * π * R * C ** exact norm_integral_le_of_norm_le_const hR hf ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R C : ℝ hR : 0 ≤ R hf : ∀ (z : ℂ), z ∈ sphere c R → ‖f z‖ ≤ C ⊢ ‖(2 * ↑π * I)⁻¹‖ = (2 * π)⁻¹ ** simp [Real.pi_pos.le] ** Qed
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circleIntegral.integral_smul ** E : Type u_1 inst✝⁵ : NormedAddCommGroup E inst✝⁴ : NormedSpace ℂ E inst✝³ : CompleteSpace E 𝕜 : Type u_2 inst✝² : IsROrC 𝕜 inst✝¹ : NormedSpace 𝕜 E inst✝ : SMulCommClass 𝕜 ℂ E a : 𝕜 f : ℂ → E c : ℂ R : ℝ ⊢ (∮ (z : ℂ) in C(c, R), a • f z) = a • ∮ (z : ℂ) in C(c, R), f z ** simp only [circleIntegral, ← smul_comm a (_ : ℂ) (_ : E), intervalIntegral.integral_smul] ** Qed
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circleIntegral.integral_sub_center_inv ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E c : ℂ R : ℝ hR : R ≠ 0 ⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹) = 2 * ↑π * I ** simp [circleIntegral, ← div_eq_mul_inv, mul_div_cancel_left _ (circleMap_ne_center hR),
intervalIntegral.integral_const I] ** Qed
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circleIntegral.integral_sub_zpow_of_undef ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E n : ℤ c w : ℂ R : ℝ hn : n < 0 hw : w ∈ sphere c |R| ⊢ (∮ (z : ℂ) in C(c, R), (z - w) ^ n) = 0 ** rcases eq_or_ne R 0 with (rfl | h0) ** case inl E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E n : ℤ c w : ℂ hn : n < 0 hw : w ∈ sphere c |0| ⊢ (∮ (z : ℂ) in C(c, 0), (z - w) ^ n) = 0 ** apply integral_radius_zero ** case inr E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E n : ℤ c w : ℂ R : ℝ hn : n < 0 hw : w ∈ sphere c |R| h0 : R ≠ 0 ⊢ (∮ (z : ℂ) in C(c, R), (z - w) ^ n) = 0 ** apply integral_undef ** case inr.hf E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E n : ℤ c w : ℂ R : ℝ hn : n < 0 hw : w ∈ sphere c |R| h0 : R ≠ 0 ⊢ ¬CircleIntegrable (fun z => (z - w) ^ n) c R ** simpa [circleIntegrable_sub_zpow_iff, *, not_or] ** Qed
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cauchyPowerSeries_apply ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ n : ℕ w : ℂ ⊢ (↑(cauchyPowerSeries f c R n) fun x => w) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z ** simp only [cauchyPowerSeries, ContinuousMultilinearMap.mkPiField_apply, Fin.prod_const,
div_eq_mul_inv, mul_pow, mul_smul, circleIntegral.integral_smul] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ n : ℕ w : ℂ ⊢ (w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z) = (2 * ↑π * I)⁻¹ • w ^ n • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z ** rw [← smul_comm (w ^ n)] ** Qed
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norm_cauchyPowerSeries_le ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ n : ℕ ⊢ ‖cauchyPowerSeries f c R n‖ = (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖ ** simp [cauchyPowerSeries, norm_smul, Real.pi_pos.le] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ n : ℕ ⊢ 0 ≤ (2 * π)⁻¹ ** simp [Real.pi_pos.le] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ n : ℕ ⊢ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖deriv (circleMap c R) θ • (circleMap c R θ - c)⁻¹ ^ n • (circleMap c R θ - c)⁻¹ • f (circleMap c R θ)‖ = (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c R x)‖))) ** simp [norm_smul, mul_left_comm |R|] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ n : ℕ ⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c R x)‖))) ≤ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n ** rcases eq_or_ne R 0 with (rfl | hR) ** case inl E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ n : ℕ ⊢ (2 * π)⁻¹ * (|0|⁻¹ ^ n * (|0| * (|0|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c 0 x)‖))) ≤ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c 0 θ)‖) * |0|⁻¹ ^ n ** cases n <;> simp [-mul_inv_rev] ** case inl.zero E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ ⊢ 0 ≤ (2 * π)⁻¹ * (2 * π * ‖f c‖) ** rw [← mul_assoc, inv_mul_cancel (Real.two_pi_pos.ne.symm), one_mul] ** case inl.zero E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ ⊢ 0 ≤ ‖f c‖ ** apply norm_nonneg ** case inr E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ n : ℕ hR : R ≠ 0 ⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0 ..2 * π, ‖f (circleMap c R x)‖))) ≤ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n ** rw [mul_inv_cancel_left₀, mul_assoc, mul_comm (|R|⁻¹ ^ n)] ** case inr.h E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ n : ℕ hR : R ≠ 0 ⊢ |R| ≠ 0 ** rwa [Ne.def, _root_.abs_eq_zero] ** Qed
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hasSum_cauchyPowerSeries_integral ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ w : ℂ hf : CircleIntegrable f c R hw : ↑Complex.abs w < R ⊢ HasSum (fun n => ↑(cauchyPowerSeries f c R n) fun x => w) ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z) ** simp only [cauchyPowerSeries_apply] ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ w : ℂ hf : CircleIntegrable f c R hw : ↑Complex.abs w < R ⊢ HasSum (fun n => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z) ** exact (hasSum_two_pi_I_cauchyPowerSeries_integral hf hw).const_smul _ ** Qed
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hasFPowerSeriesOn_cauchy_integral ** E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : CompleteSpace E f : ℂ → E c : ℂ R : ℝ≥0 hf : CircleIntegrable f c ↑R hR : 0 < R y✝ : ℂ hy : y✝ ∈ EMetric.ball 0 ↑R ⊢ ↑Complex.abs y✝ < ↑R ** simpa using hy ** Qed
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MeasureTheory.SimpleFunc.nearestPtInd_succ ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α N : ℕ x : α ⊢ ↑(nearestPtInd e (N + 1)) x = if ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x then N + 1 else ↑(nearestPtInd e N) x ** simp only [nearestPtInd, coe_piecewise, Set.piecewise] ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α N : ℕ x : α ⊢ (if x ∈ ⋂ k, ⋂ (_ : k ≤ Nat.add N 0), {x | edist (e (Nat.add N 0 + 1)) x < edist (e k) x} then ↑(const α (Nat.add N 0 + 1)) x else ↑(nearestPtInd e (Nat.add N 0)) x) = if ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x then N + 1 else ↑(nearestPtInd e N) x ** congr ** case e_c α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α N : ℕ x : α ⊢ (x ∈ ⋂ k, ⋂ (_ : k ≤ Nat.add N 0), {x | edist (e (Nat.add N 0 + 1)) x < edist (e k) x}) = ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x ** simp ** Qed
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MeasureTheory.SimpleFunc.nearestPtInd_le ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α N : ℕ x : α ⊢ ↑(nearestPtInd e N) x ≤ N ** induction' N with N ihN ** case succ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α N : ℕ ihN : ↑(nearestPtInd e N) x ≤ N ⊢ ↑(nearestPtInd e (Nat.succ N)) x ≤ Nat.succ N ** simp only [nearestPtInd_succ] ** case succ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α N : ℕ ihN : ↑(nearestPtInd e N) x ≤ N ⊢ (if ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x then N + 1 else ↑(nearestPtInd e N) x) ≤ Nat.succ N ** split_ifs ** case pos α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α N : ℕ ihN : ↑(nearestPtInd e N) x ≤ N h✝ : ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x ⊢ N + 1 ≤ Nat.succ N case neg α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α N : ℕ ihN : ↑(nearestPtInd e N) x ≤ N h✝ : ¬∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x ⊢ ↑(nearestPtInd e N) x ≤ Nat.succ N ** exacts [le_rfl, ihN.trans N.le_succ] ** case zero α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α ⊢ ↑(nearestPtInd e Nat.zero) x ≤ Nat.zero ** simp ** Qed
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MeasureTheory.SimpleFunc.edist_nearestPt_le ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k N : ℕ hk : k ≤ N ⊢ edist (↑(nearestPt e N) x) x ≤ edist (e k) x ** induction' N with N ihN generalizing k ** case zero α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k✝ N : ℕ hk✝ : k✝ ≤ N k : ℕ hk : k ≤ Nat.zero ⊢ edist (↑(nearestPt e Nat.zero) x) x ≤ edist (e k) x ** simp [nonpos_iff_eq_zero.1 hk, le_refl] ** case succ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k✝ N✝ : ℕ hk✝ : k✝ ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x k : ℕ hk : k ≤ Nat.succ N ⊢ edist (↑(nearestPt e (Nat.succ N)) x) x ≤ edist (e k) x ** simp only [nearestPt, nearestPtInd_succ, map_apply] ** case succ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k✝ N✝ : ℕ hk✝ : k✝ ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x k : ℕ hk : k ≤ Nat.succ N ⊢ edist (e (if ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x then N + 1 else ↑(nearestPtInd e N) x)) x ≤ edist (e k) x ** split_ifs with h ** case pos α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k✝ N✝ : ℕ hk✝ : k✝ ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x k : ℕ hk : k ≤ Nat.succ N h : ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x ⊢ edist (e (N + 1)) x ≤ edist (e k) x ** rcases hk.eq_or_lt with (rfl | hk) ** case pos.inl α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k N✝ : ℕ hk✝ : k ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x h : ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x hk : Nat.succ N ≤ Nat.succ N ⊢ edist (e (N + 1)) x ≤ edist (e (Nat.succ N)) x case pos.inr α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k✝ N✝ : ℕ hk✝¹ : k✝ ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x k : ℕ hk✝ : k ≤ Nat.succ N h : ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x hk : k < Nat.succ N ⊢ edist (e (N + 1)) x ≤ edist (e k) x ** exacts [le_rfl, (h k (Nat.lt_succ_iff.1 hk)).le] ** case neg α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k✝ N✝ : ℕ hk✝ : k✝ ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x k : ℕ hk : k ≤ Nat.succ N h : ¬∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x ⊢ edist (e (↑(nearestPtInd e N) x)) x ≤ edist (e k) x ** push_neg at h ** case neg α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k✝ N✝ : ℕ hk✝ : k✝ ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x k : ℕ hk : k ≤ Nat.succ N h : ∃ k, k ≤ N ∧ edist (e k) x ≤ edist (e (N + 1)) x ⊢ edist (e (↑(nearestPtInd e N) x)) x ≤ edist (e k) x ** rcases h with ⟨l, hlN, hxl⟩ ** case neg.intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k✝ N✝ : ℕ hk✝ : k✝ ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x k : ℕ hk : k ≤ Nat.succ N l : ℕ hlN : l ≤ N hxl : edist (e l) x ≤ edist (e (N + 1)) x ⊢ edist (e (↑(nearestPtInd e N) x)) x ≤ edist (e k) x ** rcases hk.eq_or_lt with (rfl | hk) ** case neg.intro.intro.inl α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k N✝ : ℕ hk✝ : k ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x l : ℕ hlN : l ≤ N hxl : edist (e l) x ≤ edist (e (N + 1)) x hk : Nat.succ N ≤ Nat.succ N ⊢ edist (e (↑(nearestPtInd e N) x)) x ≤ edist (e (Nat.succ N)) x case neg.intro.intro.inr α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α k✝ N✝ : ℕ hk✝¹ : k✝ ≤ N✝ N : ℕ ihN : ∀ {k : ℕ}, k ≤ N → edist (↑(nearestPt e N) x) x ≤ edist (e k) x k : ℕ hk✝ : k ≤ Nat.succ N l : ℕ hlN : l ≤ N hxl : edist (e l) x ≤ edist (e (N + 1)) x hk : k < Nat.succ N ⊢ edist (e (↑(nearestPtInd e N) x)) x ≤ edist (e k) x ** exacts [(ihN hlN).trans hxl, ihN (Nat.lt_succ_iff.1 hk)] ** Qed
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MeasureTheory.SimpleFunc.tendsto_nearestPt ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α hx : x ∈ closure (Set.range e) ⊢ Tendsto (fun N => ↑(nearestPt e N) x) atTop (𝓝 x) ** refine' (atTop_basis.tendsto_iff nhds_basis_eball).2 fun ε hε => _ ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α hx : x ∈ closure (Set.range e) ε : ℝ≥0∞ hε : 0 < ε ⊢ ∃ ia, True ∧ ∀ (x_1 : ℕ), x_1 ∈ Set.Ici ia → ↑(nearestPt e x_1) x ∈ ball x ε ** rcases EMetric.mem_closure_iff.1 hx ε hε with ⟨_, ⟨N, rfl⟩, hN⟩ ** case intro.intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α hx : x ∈ closure (Set.range e) ε : ℝ≥0∞ hε : 0 < ε N : ℕ hN : edist x (e N) < ε ⊢ ∃ ia, True ∧ ∀ (x_1 : ℕ), x_1 ∈ Set.Ici ia → ↑(nearestPt e x_1) x ∈ ball x ε ** rw [edist_comm] at hN ** case intro.intro.intro α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝² : MeasurableSpace α inst✝¹ : PseudoEMetricSpace α inst✝ : OpensMeasurableSpace α e : ℕ → α x : α hx : x ∈ closure (Set.range e) ε : ℝ≥0∞ hε : 0 < ε N : ℕ hN : edist (e N) x < ε ⊢ ∃ ia, True ∧ ∀ (x_1 : ℕ), x_1 ∈ Set.Ici ia → ↑(nearestPt e x_1) x ∈ ball x ε ** exact ⟨N, trivial, fun n hn => (edist_nearestPt_le e x hn).trans_lt hN⟩ ** Qed
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MeasureTheory.SimpleFunc.approxOn_mem ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace α inst✝³ : PseudoEMetricSpace α inst✝² : OpensMeasurableSpace α inst✝¹ : MeasurableSpace β f✝ f : β → α hf : Measurable f s : Set α y₀ : α h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s n : ℕ x : β ⊢ ↑(approxOn f hf s y₀ h₀ n) x ∈ s ** haveI : Nonempty s := ⟨⟨y₀, h₀⟩⟩ ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace α inst✝³ : PseudoEMetricSpace α inst✝² : OpensMeasurableSpace α inst✝¹ : MeasurableSpace β f✝ f : β → α hf : Measurable f s : Set α y₀ : α h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s n : ℕ x : β this : Nonempty ↑s ⊢ ↑(approxOn f hf s y₀ h₀ n) x ∈ s ** suffices ∀ n, (Nat.casesOn n y₀ ((↑) ∘ denseSeq s) : α) ∈ s by apply this ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace α inst✝³ : PseudoEMetricSpace α inst✝² : OpensMeasurableSpace α inst✝¹ : MeasurableSpace β f✝ f : β → α hf : Measurable f s : Set α y₀ : α h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s n : ℕ x : β this : Nonempty ↑s ⊢ ∀ (n : ℕ), Nat.casesOn n y₀ (Subtype.val ∘ denseSeq ↑s) ∈ s ** rintro (_ | n) ** case zero α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace α inst✝³ : PseudoEMetricSpace α inst✝² : OpensMeasurableSpace α inst✝¹ : MeasurableSpace β f✝ f : β → α hf : Measurable f s : Set α y₀ : α h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s n : ℕ x : β this : Nonempty ↑s ⊢ Nat.casesOn Nat.zero y₀ (Subtype.val ∘ denseSeq ↑s) ∈ s case succ α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace α inst✝³ : PseudoEMetricSpace α inst✝² : OpensMeasurableSpace α inst✝¹ : MeasurableSpace β f✝ f : β → α hf : Measurable f s : Set α y₀ : α h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s n✝ : ℕ x : β this : Nonempty ↑s n : ℕ ⊢ Nat.casesOn (Nat.succ n) y₀ (Subtype.val ∘ denseSeq ↑s) ∈ s ** exacts [h₀, Subtype.mem _] ** α : Type u_1 β : Type u_2 ι : Type u_3 E : Type u_4 F : Type u_5 𝕜 : Type u_6 inst✝⁴ : MeasurableSpace α inst✝³ : PseudoEMetricSpace α inst✝² : OpensMeasurableSpace α inst✝¹ : MeasurableSpace β f✝ f : β → α hf : Measurable f s : Set α y₀ : α h₀ : y₀ ∈ s inst✝ : SeparableSpace ↑s n : ℕ x : β this✝ : Nonempty ↑s this : ∀ (n : ℕ), Nat.casesOn n y₀ (Subtype.val ∘ denseSeq ↑s) ∈ s ⊢ ↑(approxOn f hf s y₀ h₀ n) x ∈ s ** apply this ** Qed
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aemeasurable_zero_measure ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α ⊢ AEMeasurable f ** nontriviality α ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α ✝ : Nontrivial α ⊢ AEMeasurable f ** inhabit α ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α ✝ : Nontrivial α inhabited_h : Inhabited α ⊢ AEMeasurable f ** exact ⟨fun _ => f default, measurable_const, rfl⟩ ** Qed
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aemeasurable_union_iff ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α s t : Set α ⊢ AEMeasurable f ↔ AEMeasurable f ∧ AEMeasurable f ** simp only [union_eq_iUnion, aemeasurable_iUnion_iff, Bool.forall_bool, cond, and_comm] ** Qed
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AEMeasurable.map_map_of_aemeasurable ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f ⊢ Measure.map g (Measure.map f μ) = Measure.map (g ∘ f) μ ** ext1 s hs ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f s : Set γ hs : MeasurableSet s ⊢ ↑↑(Measure.map g (Measure.map f μ)) s = ↑↑(Measure.map (g ∘ f) μ) s ** let g' := hg.mk g ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f s : Set γ hs : MeasurableSet s g' : β → γ := mk g hg ⊢ ↑↑(Measure.map g (Measure.map f μ)) s = ↑↑(Measure.map (g ∘ f) μ) s ** have A : map g (map f μ) = map g' (map f μ) := by
apply MeasureTheory.Measure.map_congr
exact hg.ae_eq_mk ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f s : Set γ hs : MeasurableSet s g' : β → γ := mk g hg A : Measure.map g (Measure.map f μ) = Measure.map g' (Measure.map f μ) ⊢ ↑↑(Measure.map g (Measure.map f μ)) s = ↑↑(Measure.map (g ∘ f) μ) s ** have B : map (g ∘ f) μ = map (g' ∘ f) μ := by
apply MeasureTheory.Measure.map_congr
exact ae_of_ae_map hf hg.ae_eq_mk ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f s : Set γ hs : MeasurableSet s g' : β → γ := mk g hg A : Measure.map g (Measure.map f μ) = Measure.map g' (Measure.map f μ) B : Measure.map (g ∘ f) μ = Measure.map (g' ∘ f) μ ⊢ ↑↑(Measure.map g (Measure.map f μ)) s = ↑↑(Measure.map (g ∘ f) μ) s ** simp only [A, B, hs, hg.measurable_mk.aemeasurable.comp_aemeasurable hf, hg.measurable_mk,
hg.measurable_mk hs, hf, map_apply, map_apply_of_aemeasurable] ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f s : Set γ hs : MeasurableSet s g' : β → γ := mk g hg A : Measure.map g (Measure.map f μ) = Measure.map g' (Measure.map f μ) B : Measure.map (g ∘ f) μ = Measure.map (g' ∘ f) μ ⊢ ↑↑μ (f ⁻¹' (mk g hg ⁻¹' s)) = ↑↑μ (mk g hg ∘ f ⁻¹' s) ** rfl ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f s : Set γ hs : MeasurableSet s g' : β → γ := mk g hg ⊢ Measure.map g (Measure.map f μ) = Measure.map g' (Measure.map f μ) ** apply MeasureTheory.Measure.map_congr ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f s : Set γ hs : MeasurableSet s g' : β → γ := mk g hg ⊢ g =ᶠ[ae (Measure.map f μ)] g' ** exact hg.ae_eq_mk ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f s : Set γ hs : MeasurableSet s g' : β → γ := mk g hg A : Measure.map g (Measure.map f μ) = Measure.map g' (Measure.map f μ) ⊢ Measure.map (g ∘ f) μ = Measure.map (g' ∘ f) μ ** apply MeasureTheory.Measure.map_congr ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g✝ : α → β μ ν : Measure α g : β → γ f : α → β hg : AEMeasurable g hf : AEMeasurable f s : Set γ hs : MeasurableSet s g' : β → γ := mk g hg A : Measure.map g (Measure.map f μ) = Measure.map g' (Measure.map f μ) ⊢ g ∘ f =ᶠ[ae μ] g' ∘ f ** exact ae_of_ae_map hf hg.ae_eq_mk ** Qed
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AEMeasurable.exists_ae_eq_range_subset ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t ⊢ ∃ g, Measurable g ∧ range g ⊆ t ∧ f =ᶠ[ae μ] g ** let s : Set α := toMeasurable μ { x | f x = H.mk f x ∧ f x ∈ t }ᶜ ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ ⊢ ∃ g, Measurable g ∧ range g ⊆ t ∧ f =ᶠ[ae μ] g ** let g : α → β := piecewise s (fun _ => h₀.some) (H.mk f) ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) ⊢ ∃ g, Measurable g ∧ range g ⊆ t ∧ f =ᶠ[ae μ] g ** refine' ⟨g, _, _, _⟩ ** case refine'_1 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) ⊢ Measurable g ** exact Measurable.piecewise (measurableSet_toMeasurable _ _) measurable_const H.measurable_mk ** case refine'_2 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) ⊢ range g ⊆ t ** rintro _ ⟨x, rfl⟩ ** case refine'_2.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) x : α ⊢ g x ∈ t ** by_cases hx : x ∈ s ** case pos ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) x : α hx : x ∈ s ⊢ g x ∈ t ** simpa [hx] using h₀.some_mem ** case neg ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) x : α hx : ¬x ∈ s ⊢ g x ∈ t ** simp only [hx, piecewise_eq_of_not_mem, not_false_iff] ** case neg ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) x : α hx : ¬x ∈ s ⊢ mk f H x ∈ t ** contrapose! hx ** case neg ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) x : α hx : ¬mk f H x ∈ t ⊢ x ∈ s ** apply subset_toMeasurable ** case neg.a ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) x : α hx : ¬mk f H x ∈ t ⊢ x ∈ {x | f x = mk f H x ∧ f x ∈ t}ᶜ ** simp (config := { contextual := true }) only [hx, mem_compl_iff, mem_setOf_eq, not_and,
not_false_iff, imp_true_iff] ** case refine'_3 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) ⊢ f =ᶠ[ae μ] g ** have A : μ (toMeasurable μ { x | f x = H.mk f x ∧ f x ∈ t }ᶜ) = 0 := by
rw [measure_toMeasurable, ← compl_mem_ae_iff, compl_compl]
exact H.ae_eq_mk.and ht ** case refine'_3 ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) A : ↑↑μ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ) = 0 ⊢ f =ᶠ[ae μ] g ** filter_upwards [compl_mem_ae_iff.2 A] with x hx ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) A : ↑↑μ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ) = 0 x : α hx : x ∈ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ)ᶜ ⊢ f x = g x ** rw [mem_compl_iff] at hx ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) A : ↑↑μ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ) = 0 x : α hx : ¬x ∈ toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ ⊢ f x = g x ** simp only [hx, piecewise_eq_of_not_mem, not_false_iff] ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) A : ↑↑μ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ) = 0 x : α hx : ¬x ∈ toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ ⊢ f x = mk f H x ** contrapose! hx ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) A : ↑↑μ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ) = 0 x : α hx : f x ≠ mk f H x ⊢ x ∈ toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ ** apply subset_toMeasurable ** case h.a ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) A : ↑↑μ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ) = 0 x : α hx : f x ≠ mk f H x ⊢ x ∈ {x | f x = mk f H x ∧ f x ∈ t}ᶜ ** simp only [hx, mem_compl_iff, mem_setOf_eq, false_and_iff, not_false_iff] ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) ⊢ ↑↑μ (toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ) = 0 ** rw [measure_toMeasurable, ← compl_mem_ae_iff, compl_compl] ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α H : AEMeasurable f t : Set β ht : ∀ᵐ (x : α) ∂μ, f x ∈ t h₀ : Set.Nonempty t s : Set α := toMeasurable μ {x | f x = mk f H x ∧ f x ∈ t}ᶜ g : α → β := piecewise s (fun x => Set.Nonempty.some h₀) (mk f H) ⊢ {x | f x = mk f H x ∧ f x ∈ t} ∈ ae μ ** exact H.ae_eq_mk.and ht ** Qed
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AEMeasurable.exists_measurable_nonneg ** ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f✝ g : α → β✝ μ ν : Measure α β : Type u_7 inst✝¹ : Preorder β inst✝ : Zero β mβ : MeasurableSpace β f : α → β hf : AEMeasurable f f_nn : ∀ᵐ (t : α) ∂μ, 0 ≤ f t ⊢ ∃ g, Measurable g ∧ 0 ≤ g ∧ f =ᶠ[ae μ] g ** obtain ⟨G, hG_meas, hG_mem, hG_ae_eq⟩ := hf.exists_ae_eq_range_subset f_nn ⟨0, le_rfl⟩ ** case intro.intro.intro ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f✝ g : α → β✝ μ ν : Measure α β : Type u_7 inst✝¹ : Preorder β inst✝ : Zero β mβ : MeasurableSpace β f : α → β hf : AEMeasurable f f_nn : ∀ᵐ (t : α) ∂μ, 0 ≤ f t G : α → β hG_meas : Measurable G hG_mem : range G ⊆ Preorder.toLE.1 0 hG_ae_eq : f =ᶠ[ae μ] G ⊢ ∃ g, Measurable g ∧ 0 ≤ g ∧ f =ᶠ[ae μ] g ** exact ⟨G, hG_meas, fun x => hG_mem (mem_range_self x), hG_ae_eq⟩ ** Qed
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AEMeasurable.subtype_mk ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α h : AEMeasurable f s : Set β hfs : ∀ (x : α), f x ∈ s ⊢ AEMeasurable (codRestrict f s hfs) ** nontriviality α ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α h : AEMeasurable f s : Set β hfs : ∀ (x : α), f x ∈ s ✝ : Nontrivial α ⊢ AEMeasurable (codRestrict f s hfs) ** inhabit α ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α h : AEMeasurable f s : Set β hfs : ∀ (x : α), f x ∈ s ✝ : Nontrivial α inhabited_h : Inhabited α ⊢ AEMeasurable (codRestrict f s hfs) ** obtain ⟨g, g_meas, hg, fg⟩ : ∃ g : α → β, Measurable g ∧ range g ⊆ s ∧ f =ᵐ[μ] g :=
h.exists_ae_eq_range_subset (eventually_of_forall hfs) ⟨_, hfs default⟩ ** case intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α h : AEMeasurable f s : Set β hfs : ∀ (x : α), f x ∈ s ✝ : Nontrivial α inhabited_h : Inhabited α g : α → β g_meas : Measurable g hg : range g ⊆ s fg : f =ᶠ[ae μ] g ⊢ AEMeasurable (codRestrict f s hfs) ** refine' ⟨codRestrict g s fun x => hg (mem_range_self _), Measurable.subtype_mk g_meas, _⟩ ** case intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α h : AEMeasurable f s : Set β hfs : ∀ (x : α), f x ∈ s ✝ : Nontrivial α inhabited_h : Inhabited α g : α → β g_meas : Measurable g hg : range g ⊆ s fg : f =ᶠ[ae μ] g ⊢ codRestrict f s hfs =ᶠ[ae μ] codRestrict g s (_ : ∀ (x : α), g x ∈ s) ** filter_upwards [fg] with x hx ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α h : AEMeasurable f s : Set β hfs : ∀ (x : α), f x ∈ s ✝ : Nontrivial α inhabited_h : Inhabited α g : α → β g_meas : Measurable g hg : range g ⊆ s fg : f =ᶠ[ae μ] g x : α hx : f x = g x ⊢ codRestrict f s hfs x = codRestrict g s (_ : ∀ (x : α), g x ∈ s) x ** simpa [Subtype.ext_iff] ** Qed
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aemeasurable_const' ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α h : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y ⊢ AEMeasurable f ** rcases eq_or_ne μ 0 with (rfl | hμ) ** case inl ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β ν : Measure α h : ∀ᵐ (x : α) (y : α) ∂0, f x = f y ⊢ AEMeasurable f ** exact aemeasurable_zero_measure ** case inr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α h : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y hμ : μ ≠ 0 ⊢ AEMeasurable f ** haveI := ae_neBot.2 hμ ** case inr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α h : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y hμ : μ ≠ 0 this : NeBot (ae μ) ⊢ AEMeasurable f ** rcases h.exists with ⟨x, hx⟩ ** case inr.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g : α → β μ ν : Measure α h : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y hμ : μ ≠ 0 this : NeBot (ae μ) x : α hx : ∀ᵐ (y : α) ∂μ, f x = f y ⊢ AEMeasurable f ** exact ⟨const α (f x), measurable_const, EventuallyEq.symm hx⟩ ** Qed
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MeasurableEmbedding.aemeasurable_map_iff ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α g : β → γ hf : MeasurableEmbedding f ⊢ AEMeasurable g ↔ AEMeasurable (g ∘ f) ** refine' ⟨fun H => H.comp_measurable hf.measurable, _⟩ ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α g : β → γ hf : MeasurableEmbedding f ⊢ AEMeasurable (g ∘ f) → AEMeasurable g ** rintro ⟨g₁, hgm₁, heq⟩ ** case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α g : β → γ hf : MeasurableEmbedding f g₁ : α → γ hgm₁ : Measurable g₁ heq : g ∘ f =ᶠ[ae μ] g₁ ⊢ AEMeasurable g ** rcases hf.exists_measurable_extend hgm₁ fun x => ⟨g x⟩ with ⟨g₂, hgm₂, rfl⟩ ** case intro.intro.intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ ν : Measure α g : β → γ hf : MeasurableEmbedding f g₂ : β → γ hgm₂ : Measurable g₂ hgm₁ : Measurable (g₂ ∘ f) heq : g ∘ f =ᶠ[ae μ] g₂ ∘ f ⊢ AEMeasurable g ** exact ⟨g₂, hgm₂, hf.ae_map_iff.2 heq⟩ ** Qed
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MeasurableEmbedding.aemeasurable_comp_iff ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ✝ ν : Measure α g : β → γ hg : MeasurableEmbedding g μ : Measure α ⊢ AEMeasurable (g ∘ f) ↔ AEMeasurable f ** refine' ⟨fun H => _, hg.measurable.comp_aemeasurable⟩ ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ✝ ν : Measure α g : β → γ hg : MeasurableEmbedding g μ : Measure α H : AEMeasurable (g ∘ f) ⊢ AEMeasurable f ** suffices AEMeasurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) μ by
rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ✝ ν : Measure α g : β → γ hg : MeasurableEmbedding g μ : Measure α H : AEMeasurable (g ∘ f) ⊢ AEMeasurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) ** exact hg.measurable_rangeSplitting.comp_aemeasurable H.subtype_mk ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f g✝ : α → β μ✝ ν : Measure α g : β → γ hg : MeasurableEmbedding g μ : Measure α H : AEMeasurable (g ∘ f) this : AEMeasurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) ⊢ AEMeasurable f ** rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this ** Qed
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aemeasurable_Ioi_of_forall_Ioc ** ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g ⊢ AEMeasurable g ** haveI : Nonempty α := ⟨x⟩ ** ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g this : Nonempty α ⊢ AEMeasurable g ** obtain ⟨u, hu_tendsto⟩ := exists_seq_tendsto (atTop : Filter α) ** case intro ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g this : Nonempty α u : ℕ → α hu_tendsto : Tendsto u atTop atTop ⊢ AEMeasurable g ** have Ioi_eq_iUnion : Ioi x = ⋃ n : ℕ, Ioc x (u n) := by
rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _]
exact fun y _ => (hu_tendsto.eventually (eventually_ge_atTop y)).exists ** case intro ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g this : Nonempty α u : ℕ → α hu_tendsto : Tendsto u atTop atTop Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n) ⊢ AEMeasurable g ** rw [Ioi_eq_iUnion, aemeasurable_iUnion_iff] ** case intro ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g this : Nonempty α u : ℕ → α hu_tendsto : Tendsto u atTop atTop Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n) ⊢ ∀ (i : ℕ), AEMeasurable g ** intro n ** case intro ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g this : Nonempty α u : ℕ → α hu_tendsto : Tendsto u atTop atTop Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n) n : ℕ ⊢ AEMeasurable g ** cases' lt_or_le x (u n) with h h ** ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g this : Nonempty α u : ℕ → α hu_tendsto : Tendsto u atTop atTop ⊢ Ioi x = ⋃ n, Ioc x (u n) ** rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _] ** ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g this : Nonempty α u : ℕ → α hu_tendsto : Tendsto u atTop atTop ⊢ ∀ (x_1 : α), x < x_1 → ∃ i, x_1 ≤ u i ** exact fun y _ => (hu_tendsto.eventually (eventually_ge_atTop y)).exists ** case intro.inl ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g this : Nonempty α u : ℕ → α hu_tendsto : Tendsto u atTop atTop Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n) n : ℕ h : x < u n ⊢ AEMeasurable g ** exact g_meas (u n) h ** case intro.inr ι : Type u_1 α : Type u_2 β✝ : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝⁴ : MeasurableSpace β✝ inst✝³ : MeasurableSpace γ inst✝² : MeasurableSpace δ f g✝ : α → β✝ μ ν : Measure α β : Type u_7 mβ : MeasurableSpace β inst✝¹ : LinearOrder α inst✝ : IsCountablyGenerated atTop x : α g : α → β g_meas : ∀ (t : α), t > x → AEMeasurable g this : Nonempty α u : ℕ → α hu_tendsto : Tendsto u atTop atTop Ioi_eq_iUnion : Ioi x = ⋃ n, Ioc x (u n) n : ℕ h : u n ≤ x ⊢ AEMeasurable g ** exact aemeasurable_zero_measure ** Qed
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aemeasurable_indicator_iff ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s : Set α hs : MeasurableSet s ⊢ AEMeasurable (indicator s f) ↔ AEMeasurable f ** constructor ** case mp ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s : Set α hs : MeasurableSet s ⊢ AEMeasurable (indicator s f) → AEMeasurable f ** intro h ** case mp ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s : Set α hs : MeasurableSet s h : AEMeasurable (indicator s f) ⊢ AEMeasurable f ** exact (h.mono_measure Measure.restrict_le_self).congr (indicator_ae_eq_restrict hs) ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s : Set α hs : MeasurableSet s ⊢ AEMeasurable f → AEMeasurable (indicator s f) ** intro h ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s : Set α hs : MeasurableSet s h : AEMeasurable f ⊢ AEMeasurable (indicator s f) ** refine' ⟨indicator s (h.mk f), h.measurable_mk.indicator hs, _⟩ ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s : Set α hs : MeasurableSet s h : AEMeasurable f ⊢ indicator s f =ᶠ[ae μ] indicator s (AEMeasurable.mk f h) ** have A : s.indicator f =ᵐ[μ.restrict s] s.indicator (AEMeasurable.mk f h) :=
(indicator_ae_eq_restrict hs).trans (h.ae_eq_mk.trans <| (indicator_ae_eq_restrict hs).symm) ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s : Set α hs : MeasurableSet s h : AEMeasurable f A : indicator s f =ᶠ[ae (Measure.restrict μ s)] indicator s (AEMeasurable.mk f h) ⊢ indicator s f =ᶠ[ae μ] indicator s (AEMeasurable.mk f h) ** have B : s.indicator f =ᵐ[μ.restrict sᶜ] s.indicator (AEMeasurable.mk f h) :=
(indicator_ae_eq_restrict_compl hs).trans (indicator_ae_eq_restrict_compl hs).symm ** case mpr ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s : Set α hs : MeasurableSet s h : AEMeasurable f A : indicator s f =ᶠ[ae (Measure.restrict μ s)] indicator s (AEMeasurable.mk f h) B : indicator s f =ᶠ[ae (Measure.restrict μ sᶜ)] indicator s (AEMeasurable.mk f h) ⊢ indicator s f =ᶠ[ae μ] indicator s (AEMeasurable.mk f h) ** exact ae_of_ae_restrict_of_ae_restrict_compl _ A B ** Qed
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aemeasurable_indicator_iff₀ ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s : Set α hs : NullMeasurableSet s ⊢ AEMeasurable (indicator s f) ↔ AEMeasurable f ** rcases hs with ⟨t, ht, hst⟩ ** case intro.intro ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝³ : MeasurableSpace β inst✝² : MeasurableSpace γ inst✝¹ : MeasurableSpace δ f g : α → β μ ν : Measure α inst✝ : Zero β s t : Set α ht : MeasurableSet t hst : s =ᶠ[ae μ] t ⊢ AEMeasurable (indicator s f) ↔ AEMeasurable f ** rw [← aemeasurable_congr (indicator_ae_eq_of_ae_eq_set hst.symm), aemeasurable_indicator_iff ht,
restrict_congr_set hst] ** Qed
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MeasureTheory.Measure.restrict_map_of_aemeasurable ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g : α → β μ ν : Measure α f : α → δ hf : AEMeasurable f s : Set δ hs : MeasurableSet s ⊢ restrict (map f μ) s = restrict (map (AEMeasurable.mk f hf) μ) s ** congr 1 ** case e_μ ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g : α → β μ ν : Measure α f : α → δ hf : AEMeasurable f s : Set δ hs : MeasurableSet s ⊢ map f μ = map (AEMeasurable.mk f hf) μ ** apply Measure.map_congr hf.ae_eq_mk ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g : α → β μ ν : Measure α f : α → δ hf : AEMeasurable f s : Set δ hs : MeasurableSet s ⊢ map f (restrict μ (AEMeasurable.mk f hf ⁻¹' s)) = map f (restrict μ (f ⁻¹' s)) ** apply congr_arg ** case h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g : α → β μ ν : Measure α f : α → δ hf : AEMeasurable f s : Set δ hs : MeasurableSet s ⊢ restrict μ (AEMeasurable.mk f hf ⁻¹' s) = restrict μ (f ⁻¹' s) ** ext1 t ht ** case h.h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g : α → β μ ν : Measure α f : α → δ hf : AEMeasurable f s : Set δ hs : MeasurableSet s t : Set α ht : MeasurableSet t ⊢ ↑↑(restrict μ (AEMeasurable.mk f hf ⁻¹' s)) t = ↑↑(restrict μ (f ⁻¹' s)) t ** simp only [ht, Measure.restrict_apply] ** case h.h ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g : α → β μ ν : Measure α f : α → δ hf : AEMeasurable f s : Set δ hs : MeasurableSet s t : Set α ht : MeasurableSet t ⊢ ↑↑μ (t ∩ AEMeasurable.mk f hf ⁻¹' s) = ↑↑μ (t ∩ f ⁻¹' s) ** apply measure_congr ** case h.h.H ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g : α → β μ ν : Measure α f : α → δ hf : AEMeasurable f s : Set δ hs : MeasurableSet s t : Set α ht : MeasurableSet t ⊢ t ∩ AEMeasurable.mk f hf ⁻¹' s =ᶠ[ae μ] t ∩ f ⁻¹' s ** apply (EventuallyEq.refl _ _).inter (hf.ae_eq_mk.symm.preimage s) ** Qed
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MeasureTheory.Measure.map_mono_of_aemeasurable ** ι : Type u_1 α : Type u_2 β : Type u_3 γ : Type u_4 δ : Type u_5 R : Type u_6 m0 : MeasurableSpace α inst✝² : MeasurableSpace β inst✝¹ : MeasurableSpace γ inst✝ : MeasurableSpace δ f✝ g : α → β μ ν : Measure α f : α → δ h : μ ≤ ν hf : AEMeasurable f s : Set δ hs : MeasurableSet s ⊢ ↑↑(map f μ) s ≤ ↑↑(map f ν) s ** simpa [hf, hs, hf.mono_measure h] using Measure.le_iff'.1 h (f ⁻¹' s) ** Qed
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MeasureTheory.Measure.toOuterMeasure_injective ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α x✝¹ x✝ : Measure α m₁ : OuterMeasure α u₁ : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), MeasurableSet (f i)) → Pairwise (Disjoint on f) → ↑m₁ (⋃ i, f i) = ∑' (i : ℕ), ↑m₁ (f i) h₁ : OuterMeasure.trim m₁ = m₁ m₂ : OuterMeasure α _u₂ : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), MeasurableSet (f i)) → Pairwise (Disjoint on f) → ↑m₂ (⋃ i, f i) = ∑' (i : ℕ), ↑m₂ (f i) _h₂ : OuterMeasure.trim m₂ = m₂ _h : ↑{ toOuterMeasure := m₁, m_iUnion := u₁, trimmed := h₁ } = ↑{ toOuterMeasure := m₂, m_iUnion := _u₂, trimmed := _h₂ } ⊢ { toOuterMeasure := m₁, m_iUnion := u₁, trimmed := h₁ } = { toOuterMeasure := m₂, m_iUnion := _u₂, trimmed := _h₂ } ** congr ** Qed
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MeasureTheory.Measure.ext_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α ⊢ μ₁ = μ₂ → ∀ (s : Set α), MeasurableSet s → ↑↑μ₁ s = ↑↑μ₂ s ** rintro rfl s _hs ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ : Measure α s✝ s₁ s₂ t s : Set α _hs : MeasurableSet s ⊢ ↑↑μ₁ s = ↑↑μ₁ s ** rfl ** Qed
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MeasureTheory.Measure.ext_iff' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α ⊢ μ₁ = μ₂ → ∀ (s : Set α), ↑↑μ₁ s = ↑↑μ₂ s ** rintro rfl s ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ : Measure α s✝ s₁ s₂ t s : Set α ⊢ ↑↑μ₁ s = ↑↑μ₁ s ** rfl ** Qed
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MeasureTheory.measure_eq_trim ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t s : Set α ⊢ ↑↑μ s = ↑(OuterMeasure.trim ↑μ) s ** rw [μ.trimmed] ** Qed
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MeasureTheory.measure_eq_iInf ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t s : Set α ⊢ ↑↑μ s = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑↑μ t ** rw [measure_eq_trim, OuterMeasure.trim_eq_iInf] ** Qed
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MeasureTheory.measure_eq_iInf' ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α ⊢ ↑↑μ s = ⨅ t, ↑↑μ ↑t ** simp_rw [iInf_subtype, iInf_and, ← measure_eq_iInf] ** Qed
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MeasureTheory.measure_eq_extend ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α hs : MeasurableSet s ⊢ ↑↑μ s = extend (fun t _ht => ↑↑μ t) s ** rw [extend_eq] ** case h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α hs : MeasurableSet s ⊢ MeasurableSet s ** exact hs ** Qed
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MeasureTheory.exists_measurable_superset ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s ** simpa only [← measure_eq_trim] using μ.toOuterMeasure.exists_measurable_superset_eq_trim s ** Qed
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MeasureTheory.exists_measurable_superset_forall_eq ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι✝ : Type u_5 inst✝¹ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α ι : Sort u_6 inst✝ : Countable ι μ : ι → Measure α s : Set α ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑↑(μ i) t = ↑↑(μ i) s ** simpa only [← measure_eq_trim] using
OuterMeasure.exists_measurable_superset_forall_eq_trim (fun i => (μ i).toOuterMeasure) s ** Qed
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MeasureTheory.exists_measurable_superset₂ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ ν : Measure α s : Set α ⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s ∧ ↑↑ν t = ↑↑ν s ** simpa only [Bool.forall_bool.trans and_comm] using
exists_measurable_superset_forall_eq (fun b => cond b μ ν) s ** Qed
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MeasureTheory.measure_biUnion_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Set β hs : Set.Countable s f : β → Set α ⊢ ↑↑μ (⋃ b ∈ s, f b) ≤ ∑' (p : ↑s), ↑↑μ (f ↑p) ** haveI := hs.to_subtype ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Set β hs : Set.Countable s f : β → Set α this : Countable ↑s ⊢ ↑↑μ (⋃ b ∈ s, f b) ≤ ∑' (p : ↑s), ↑↑μ (f ↑p) ** rw [biUnion_eq_iUnion] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Set β hs : Set.Countable s f : β → Set α this : Countable ↑s ⊢ ↑↑μ (⋃ x, f ↑x) ≤ ∑' (p : ↑s), ↑↑μ (f ↑p) ** apply measure_iUnion_le ** Qed
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MeasureTheory.measure_biUnion_finset_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Finset β f : β → Set α ⊢ ↑↑μ (⋃ b ∈ s, f b) ≤ ∑ p in s, ↑↑μ (f p) ** rw [← Finset.sum_attach, Finset.attach_eq_univ, ← tsum_fintype] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Finset β f : β → Set α ⊢ ↑↑μ (⋃ b ∈ s, f b) ≤ ∑' (b : { x // x ∈ s }), ↑↑μ (f ↑b) ** exact measure_biUnion_le s.countable_toSet f ** Qed
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MeasureTheory.measure_iUnion_fintype_le ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝¹ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α inst✝ : Fintype β f : β → Set α ⊢ ↑↑μ (⋃ b, f b) ≤ ∑ p : β, ↑↑μ (f p) ** convert measure_biUnion_finset_le Finset.univ f ** case h.e'_3.h.e'_3.h.e'_3.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝¹ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α inst✝ : Fintype β f : β → Set α x✝ : β ⊢ f x✝ = ⋃ (_ : x✝ ∈ Finset.univ), f x✝ ** simp ** Qed
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MeasureTheory.measure_biUnion_lt_top ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Set β f : β → Set α hs : Set.Finite s hfin : ∀ (i : β), i ∈ s → ↑↑μ (f i) ≠ ⊤ ⊢ ↑↑μ (⋃ i ∈ s, f i) < ⊤ ** convert (measure_biUnion_finset_le hs.toFinset f).trans_lt _ using 3 ** case convert_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Set β f : β → Set α hs : Set.Finite s hfin : ∀ (i : β), i ∈ s → ↑↑μ (f i) ≠ ⊤ ⊢ ∑ p in Finite.toFinset hs, ↑↑μ (f p) < ⊤ ** apply ENNReal.sum_lt_top ** case convert_3.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Set β f : β → Set α hs : Set.Finite s hfin : ∀ (i : β), i ∈ s → ↑↑μ (f i) ≠ ⊤ ⊢ ∀ (a : β), a ∈ Finite.toFinset hs → ↑↑μ (f a) ≠ ⊤ ** simpa only [Finite.mem_toFinset] ** case h.e'_3.h.e'_3.h.e'_3 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Set β f : β → Set α hs : Set.Finite s hfin : ∀ (i : β), i ∈ s → ↑↑μ (f i) ≠ ⊤ ⊢ (fun i => ⋃ (_ : i ∈ s), f i) = fun b => ⋃ (_ : b ∈ Finite.toFinset hs), f b ** ext ** case h.e'_3.h.e'_3.h.e'_3.h.h α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α s : Set β f : β → Set α hs : Set.Finite s hfin : ∀ (i : β), i ∈ s → ↑↑μ (f i) ≠ ⊤ x✝¹ : β x✝ : α ⊢ x✝ ∈ ⋃ (_ : x✝¹ ∈ s), f x✝¹ ↔ x✝ ∈ ⋃ (_ : x✝¹ ∈ Finite.toFinset hs), f x✝¹ ** rw [Finite.mem_toFinset] ** Qed
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MeasureTheory.measure_union_lt_top_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α ⊢ ↑↑μ (s ∪ t) < ⊤ ↔ ↑↑μ s < ⊤ ∧ ↑↑μ t < ⊤ ** refine' ⟨fun h => ⟨_, _⟩, fun h => measure_union_lt_top h.1 h.2⟩ ** case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : ↑↑μ (s ∪ t) < ⊤ ⊢ ↑↑μ s < ⊤ ** exact (measure_mono (Set.subset_union_left s t)).trans_lt h ** case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : ↑↑μ (s ∪ t) < ⊤ ⊢ ↑↑μ t < ⊤ ** exact (measure_mono (Set.subset_union_right s t)).trans_lt h ** Qed
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MeasureTheory.exists_measure_pos_of_not_measure_iUnion_null ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝¹ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α inst✝ : Countable β s : β → Set α hs : ↑↑μ (⋃ n, s n) ≠ 0 ⊢ ∃ n, 0 < ↑↑μ (s n) ** contrapose! hs ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝¹ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α inst✝ : Countable β s : β → Set α hs : ∀ (n : β), ↑↑μ (s n) ≤ 0 ⊢ ↑↑μ (⋃ n, s n) = 0 ** exact measure_iUnion_null fun n => nonpos_iff_eq_zero.1 (hs n) ** Qed
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MeasureTheory.compl_mem_ae_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t s : Set α ⊢ sᶜ ∈ Measure.ae μ ↔ ↑↑μ s = 0 ** simp only [mem_ae_iff, compl_compl] ** Qed
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MeasureTheory.all_ae_of ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι✝ : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α ι : Sort u_6 p : α → ι → Prop hp : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), p a i i : ι ⊢ ∀ᵐ (a : α) ∂μ, p a i ** filter_upwards [hp] with a ha using ha i ** Qed
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MeasureTheory.ae_le_of_ae_lt ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α f g : α → ℝ≥0∞ h : ∀ᵐ (x : α) ∂μ, f x < g x ⊢ f ≤ᵐ[μ] g ** rw [Filter.EventuallyLE, ae_iff] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α f g : α → ℝ≥0∞ h : ∀ᵐ (x : α) ∂μ, f x < g x ⊢ ↑↑μ {a | ¬f a ≤ g a} = 0 ** rw [ae_iff] at h ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α f g : α → ℝ≥0∞ h : ↑↑μ {a | ¬f a < g a} = 0 ⊢ ↑↑μ {a | ¬f a ≤ g a} = 0 ** refine' measure_mono_null (fun x hx => _) h ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α f g : α → ℝ≥0∞ h : ↑↑μ {a | ¬f a < g a} = 0 x : α hx : x ∈ {a | ¬f a ≤ g a} ⊢ x ∈ {a | ¬f a < g a} ** exact not_lt.2 (le_of_lt (not_le.1 hx)) ** Qed
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MeasureTheory.ae_eq_empty ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α ⊢ (∀ᵐ (x : α) ∂μ, ¬x ∈ s) ↔ ↑↑μ s = 0 ** simp only [ae_iff, Classical.not_not, setOf_mem_eq] ** Qed
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MeasureTheory.ae_le_set ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α ⊢ (∀ᵐ (x : α) ∂μ, x ∈ s → x ∈ t) ↔ ↑↑μ (s \ t) = 0 ** simp [ae_iff] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α ⊢ ↑↑μ {a | a ∈ s ∧ ¬a ∈ t} = 0 ↔ ↑↑μ (s \ t) = 0 ** rfl ** Qed
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MeasureTheory.union_ae_eq_right ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α ⊢ s ∪ t =ᵐ[μ] t ↔ ↑↑μ (s \ t) = 0 ** simp [eventuallyLE_antisymm_iff, ae_le_set, union_diff_right,
diff_eq_empty.2 (Set.subset_union_right _ _)] ** Qed
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MeasureTheory.diff_ae_eq_self ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α ⊢ s \ t =ᵐ[μ] s ↔ ↑↑μ (s ∩ t) = 0 ** simp [eventuallyLE_antisymm_iff, ae_le_set, diff_diff_right, diff_diff,
diff_eq_empty.2 (Set.subset_union_right _ _)] ** Qed
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MeasureTheory.ae_eq_set ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t✝ s t : Set α ⊢ s =ᵐ[μ] t ↔ ↑↑μ (s \ t) = 0 ∧ ↑↑μ (t \ s) = 0 ** simp [eventuallyLE_antisymm_iff, ae_le_set] ** Qed
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MeasureTheory.measure_symmDiff_eq_zero_iff ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t✝ s t : Set α ⊢ ↑↑μ (s ∆ t) = 0 ↔ s =ᵐ[μ] t ** simp [ae_eq_set, symmDiff_def] ** Qed
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MeasureTheory.ae_eq_set_compl_compl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t✝ s t : Set α ⊢ sᶜ =ᵐ[μ] tᶜ ↔ s =ᵐ[μ] t ** simp only [← measure_symmDiff_eq_zero_iff, compl_symmDiff_compl] ** Qed
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MeasureTheory.ae_eq_set_compl ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t✝ s t : Set α ⊢ sᶜ =ᵐ[μ] t ↔ s =ᵐ[μ] tᶜ ** rw [← ae_eq_set_compl_compl, compl_compl] ** Qed
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MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_left ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : s =ᵐ[μ] univ ⊢ s ∪ t =ᵐ[μ] univ ** convert ae_eq_set_union h (ae_eq_refl t) ** case h.e'_5 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : s =ᵐ[μ] univ ⊢ univ = univ ∪ t ** rw [univ_union] ** Qed
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MeasureTheory.union_ae_eq_right_of_ae_eq_empty ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : s =ᵐ[μ] ∅ ⊢ s ∪ t =ᵐ[μ] t ** convert ae_eq_set_union h (ae_eq_refl t) ** case h.e'_5 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : s =ᵐ[μ] ∅ ⊢ t = ∅ ∪ t ** rw [empty_union] ** Qed
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MeasureTheory.union_ae_eq_left_of_ae_eq_empty ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : t =ᵐ[μ] ∅ ⊢ s ∪ t =ᵐ[μ] s ** convert ae_eq_set_union (ae_eq_refl s) h ** case h.e'_5 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : t =ᵐ[μ] ∅ ⊢ s = s ∪ ∅ ** rw [union_empty] ** Qed
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MeasureTheory.inter_ae_eq_right_of_ae_eq_univ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : s =ᵐ[μ] univ ⊢ s ∩ t =ᵐ[μ] t ** convert ae_eq_set_inter h (ae_eq_refl t) ** case h.e'_5 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : s =ᵐ[μ] univ ⊢ t = univ ∩ t ** rw [univ_inter] ** Qed
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MeasureTheory.inter_ae_eq_left_of_ae_eq_univ ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : t =ᵐ[μ] univ ⊢ s ∩ t =ᵐ[μ] s ** convert ae_eq_set_inter (ae_eq_refl s) h ** case h.e'_5 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : t =ᵐ[μ] univ ⊢ s = s ∩ univ ** rw [inter_univ] ** Qed
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MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_left ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : s =ᵐ[μ] ∅ ⊢ s ∩ t =ᵐ[μ] ∅ ** convert ae_eq_set_inter h (ae_eq_refl t) ** case h.e'_5 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : s =ᵐ[μ] ∅ ⊢ ∅ = ∅ ∩ t ** rw [empty_inter] ** Qed
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MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_right ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : t =ᵐ[μ] ∅ ⊢ s ∩ t =ᵐ[μ] ∅ ** convert ae_eq_set_inter (ae_eq_refl s) h ** case h.e'_5 α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α h : t =ᵐ[μ] ∅ ⊢ ∅ = s ∩ ∅ ** rw [inter_empty] ** Qed
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MeasurableSpace.ae_induction_on_inter ** α : Type u_1 β✝ : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝¹ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α β : Type u_6 inst✝ : MeasurableSpace β μ : Measure β C : β → Set α → Prop s : Set (Set α) m : MeasurableSpace α h_eq : m = generateFrom s h_inter : IsPiSystem s h_empty : ∀ᵐ (x : β) ∂μ, C x ∅ h_basic : ∀ᵐ (x : β) ∂μ, ∀ (t : Set α), t ∈ s → C x t h_compl : ∀ᵐ (x : β) ∂μ, ∀ (t : Set α), MeasurableSet t → C x t → C x tᶜ h_union : ∀ᵐ (x : β) ∂μ, ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C x (f i)) → C x (⋃ i, f i) ⊢ ∀ᵐ (x : β) ∂μ, ∀ ⦃t : Set α⦄, MeasurableSet t → C x t ** filter_upwards [h_empty, h_basic, h_compl, h_union] with x hx_empty hx_basic hx_compl hx_union
using MeasurableSpace.induction_on_inter h_eq h_inter hx_empty hx_basic hx_compl hx_union ** Qed
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Set.mulIndicator_ae_eq_one ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝¹ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α M : Type u_6 inst✝ : One M f : α → M s : Set α ⊢ mulIndicator s f =ᵐ[μ] 1 ↔ ↑↑μ (s ∩ mulSupport f) = 0 ** simp [EventuallyEq, eventually_iff, Measure.ae, compl_setOf] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝¹ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α M : Type u_6 inst✝ : One M f : α → M s : Set α ⊢ ↑↑μ {a | a ∈ s ∧ ¬f a = 1} = 0 ↔ ↑↑μ (s ∩ mulSupport f) = 0 ** rfl ** Qed
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MeasureTheory.measure_mono_ae ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α H : s ≤ᵐ[μ] t ⊢ ↑↑μ (s ∪ t) = ↑↑μ (t ∪ s \ t) ** rw [union_diff_self, Set.union_comm] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s s₁ s₂ t : Set α H : s ≤ᵐ[μ] t ⊢ ↑↑μ t + ↑↑μ (s \ t) = ↑↑μ t ** rw [ae_le_set.1 H, add_zero] ** Qed
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MeasureTheory.subset_toMeasurable ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α ⊢ s ⊆ toMeasurable μ s ** rw [toMeasurable_def] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α ⊢ s ⊆ if h : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s then Exists.choose h else if h' : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) then Exists.choose h' else Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s) ** split_ifs with hs h's ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α hs : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s ⊢ s ⊆ Exists.choose hs case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α hs : ¬∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s h's : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) ⊢ s ⊆ Exists.choose h's case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α hs : ¬∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s h's : ¬∃ t, t ⊇ s ∧ MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) ⊢ s ⊆ Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s) ** exacts [hs.choose_spec.1, h's.choose_spec.1, (exists_measurable_superset μ s).choose_spec.1] ** Qed
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MeasureTheory.measurableSet_toMeasurable ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α ⊢ MeasurableSet (toMeasurable μ s) ** rw [toMeasurable_def] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α ⊢ MeasurableSet (if h : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s then Exists.choose h else if h' : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) then Exists.choose h' else Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s)) ** split_ifs with hs h's ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α hs : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s ⊢ MeasurableSet (Exists.choose hs) case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α hs : ¬∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s h's : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) ⊢ MeasurableSet (Exists.choose h's) case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ✝ μ₁ μ₂ : Measure α s✝ s₁ s₂ t : Set α μ : Measure α s : Set α hs : ¬∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s h's : ¬∃ t, t ⊇ s ∧ MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) ⊢ MeasurableSet (Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s)) ** exacts [hs.choose_spec.2.1, h's.choose_spec.2.1,
(exists_measurable_superset μ s).choose_spec.2.1] ** Qed
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MeasureTheory.measure_toMeasurable ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t s : Set α ⊢ ↑↑μ (toMeasurable μ s) = ↑↑μ s ** rw [toMeasurable_def] ** α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t s : Set α ⊢ ↑↑μ (if h : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s then Exists.choose h else if h' : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) then Exists.choose h' else Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s)) = ↑↑μ s ** split_ifs with hs h's ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t s : Set α hs : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s ⊢ ↑↑μ (Exists.choose hs) = ↑↑μ s ** exact measure_congr hs.choose_spec.2.2 ** case pos α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t s : Set α hs : ¬∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s h's : ∃ t, t ⊇ s ∧ MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) ⊢ ↑↑μ (Exists.choose h's) = ↑↑μ s ** simpa only [inter_univ] using h's.choose_spec.2.2 univ MeasurableSet.univ ** case neg α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 ι : Type u_5 inst✝ : MeasurableSpace α μ μ₁ μ₂ : Measure α s✝ s₁ s₂ t s : Set α hs : ¬∃ t, t ⊇ s ∧ MeasurableSet t ∧ t =ᵐ[μ] s h's : ¬∃ t, t ⊇ s ∧ MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) ⊢ ↑↑μ (Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s)) = ↑↑μ s ** exact (exists_measurable_superset μ s).choose_spec.2.2 ** Qed
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MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b ⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a ** set e : ℝ ≃L[ℝ] ℝ¹ := (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ).symm ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) ⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a ** have e_symm : ∀ x, e.symm x = x 0 := fun x => rfl ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 ⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a ** set F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight (1 : ℝ →L[ℝ] ℝ) (f' x) ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) ⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a ** have hF' : ∀ x y, F' x y = y • f' x := fun x y => rfl ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x ⊢ ∫ (x : ℝ) in a..b, f' x = ∫ (x : ℝ) in Set.Icc a b, f' x ** rw [intervalIntegral.integral_of_le hle, set_integral_congr_set_ae Ioc_ae_eq_Icc] ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x ⊢ ∫ (x : ℝ) in Set.Icc a b, f' x = ∑ i : Fin 1, ((∫ (x : Fin 0 → ℝ) in Set.Icc (↑e a ∘ Fin.succAbove i) (↑e b ∘ Fin.succAbove i), f (↑(ContinuousLinearEquiv.symm e) (Fin.insertNth i (↑e b i) x))) - ∫ (x : Fin 0 → ℝ) in Set.Icc (↑e a ∘ Fin.succAbove i) (↑e b ∘ Fin.succAbove i), f (↑(ContinuousLinearEquiv.symm e) (Fin.insertNth i (↑e a i) x))) ** simp only [← interior_Icc] at Hd ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x Hd : ∀ (x : ℝ), x ∈ interior (Set.Icc a b) \ s → HasDerivAt f (f' x) x ⊢ ∫ (x : ℝ) in Set.Icc a b, f' x = ∑ i : Fin 1, ((∫ (x : Fin 0 → ℝ) in Set.Icc (↑e a ∘ Fin.succAbove i) (↑e b ∘ Fin.succAbove i), f (↑(ContinuousLinearEquiv.symm e) (Fin.insertNth i (↑e b i) x))) - ∫ (x : Fin 0 → ℝ) in Set.Icc (↑e a ∘ Fin.succAbove i) (↑e b ∘ Fin.succAbove i), f (↑(ContinuousLinearEquiv.symm e) (Fin.insertNth i (↑e a i) x))) ** refine'
integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv e _ _ (fun _ => f)
(fun _ => F') s hs a b hle (fun _ => Hc) (fun x hx _ => Hd x hx) _ _ _ ** case refine'_1 E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x Hd : ∀ (x : ℝ), x ∈ interior (Set.Icc a b) \ s → HasDerivAt f (f' x) x ⊢ ∀ (x y : ℝ), ↑e x ≤ ↑e y ↔ x ≤ y ** exact fun x y => (OrderIso.funUnique (Fin 1) ℝ).symm.le_iff_le ** case refine'_2 E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x Hd : ∀ (x : ℝ), x ∈ interior (Set.Icc a b) \ s → HasDerivAt f (f' x) x ⊢ MeasurePreserving ↑e ** exact (volume_preserving_funUnique (Fin 1) ℝ).symm _ ** case refine'_3 E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x Hd : ∀ (x : ℝ), x ∈ interior (Set.Icc a b) \ s → HasDerivAt f (f' x) x ⊢ ∀ (x : ℝ), f' x = ∑ i : Fin (0 + 1), ↑((fun x => F') i x) (↑(ContinuousLinearEquiv.symm e) (Pi.single i 1)) ** intro x ** case refine'_3 E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x Hd : ∀ (x : ℝ), x ∈ interior (Set.Icc a b) \ s → HasDerivAt f (f' x) x x : ℝ ⊢ f' x = ∑ i : Fin (0 + 1), ↑((fun x => F') i x) (↑(ContinuousLinearEquiv.symm e) (Pi.single i 1)) ** rw [Fin.sum_univ_one, hF', e_symm, Pi.single_eq_same, one_smul] ** case refine'_4 E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x Hd : ∀ (x : ℝ), x ∈ interior (Set.Icc a b) \ s → HasDerivAt f (f' x) x ⊢ IntegrableOn (fun x => f' x) (Set.Icc a b) ** rw [intervalIntegrable_iff_integrable_Ioc_of_le hle] at Hi ** case refine'_4 E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hi : IntegrableOn f' (Set.Ioc a b) e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x Hd : ∀ (x : ℝ), x ∈ interior (Set.Icc a b) \ s → HasDerivAt f (f' x) x ⊢ IntegrableOn (fun x => f' x) (Set.Icc a b) ** exact Hi.congr_set_ae Ioc_ae_eq_Icc.symm ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x ⊢ ∑ i : Fin 1, ((∫ (x : Fin 0 → ℝ) in Set.Icc (↑e a ∘ Fin.succAbove i) (↑e b ∘ Fin.succAbove i), f (↑(ContinuousLinearEquiv.symm e) (Fin.insertNth i (↑e b i) x))) - ∫ (x : Fin 0 → ℝ) in Set.Icc (↑e a ∘ Fin.succAbove i) (↑e b ∘ Fin.succAbove i), f (↑(ContinuousLinearEquiv.symm e) (Fin.insertNth i (↑e a i) x))) = f b - f a ** simp only [Fin.sum_univ_one, e_symm] ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x ⊢ (∫ (x : Fin 0 → ℝ) in Set.Icc (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) a ∘ Fin.succAbove 0) (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) b ∘ Fin.succAbove 0), f (Fin.insertNth 0 (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) b 0) x 0)) - ∫ (x : Fin 0 → ℝ) in Set.Icc (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) a ∘ Fin.succAbove 0) (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) b ∘ Fin.succAbove 0), f (Fin.insertNth 0 (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) a 0) x 0) = f b - f a ** have : ∀ c : ℝ, const (Fin 0) c = isEmptyElim := fun c => Subsingleton.elim _ _ ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ hle : a ≤ b s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b e : ℝ ≃L[ℝ] Fin 1 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ) e_symm : ∀ (x : Fin 1 → ℝ), ↑(ContinuousLinearEquiv.symm e) x = x 0 F' : ℝ → ℝ →L[ℝ] E := fun x => smulRight 1 (f' x) hF' : ∀ (x y : ℝ), ↑(F' x) y = y • f' x this : ∀ (c : ℝ), const (Fin 0) c = fun a => isEmptyElim a ⊢ (∫ (x : Fin 0 → ℝ) in Set.Icc (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) a ∘ Fin.succAbove 0) (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) b ∘ Fin.succAbove 0), f (Fin.insertNth 0 (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) b 0) x 0)) - ∫ (x : Fin 0 → ℝ) in Set.Icc (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) a ∘ Fin.succAbove 0) (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) b ∘ Fin.succAbove 0), f (Fin.insertNth 0 (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.funUnique (Fin 1) ℝ ℝ)) a 0) x 0) = f b - f a ** simp [this, volume_pi, Measure.pi_of_empty fun _ : Fin 0 => volume] ** Qed
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MeasureTheory.integral_eq_of_has_deriv_within_at_off_countable ** E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f [[a, b]] Hd : ∀ (x : ℝ), x ∈ Set.Ioo (min a b) (max a b) \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b ⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a ** cases' le_total a b with hab hab ** case inl E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f [[a, b]] Hd : ∀ (x : ℝ), x ∈ Set.Ioo (min a b) (max a b) \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b hab : a ≤ b ⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a ** simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at * ** case inl E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ s : Set ℝ hs : Set.Countable s Hi : IntervalIntegrable f' volume a b hab : a ≤ b Hc : ContinuousOn f (Set.Icc a b) Hd : ∀ (x : ℝ), x ∈ Set.Ioo a b \ s → HasDerivAt f (f' x) x ⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a ** exact integral_eq_of_hasDerivWithinAt_off_countable_of_le f f' hab hs Hc Hd Hi ** case inr E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ s : Set ℝ hs : Set.Countable s Hc : ContinuousOn f [[a, b]] Hd : ∀ (x : ℝ), x ∈ Set.Ioo (min a b) (max a b) \ s → HasDerivAt f (f' x) x Hi : IntervalIntegrable f' volume a b hab : b ≤ a ⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a ** simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at * ** case inr E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ s : Set ℝ hs : Set.Countable s Hi : IntervalIntegrable f' volume a b hab : b ≤ a Hc : ContinuousOn f (Set.Icc b a) Hd : ∀ (x : ℝ), x ∈ Set.Ioo b a \ s → HasDerivAt f (f' x) x ⊢ ∫ (x : ℝ) in a..b, f' x = f b - f a ** rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub] ** case inr E : Type u inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℝ E inst✝ : CompleteSpace E f f' : ℝ → E a b : ℝ s : Set ℝ hs : Set.Countable s Hi : IntervalIntegrable f' volume a b hab : b ≤ a Hc : ContinuousOn f (Set.Icc b a) Hd : ∀ (x : ℝ), x ∈ Set.Ioo b a \ s → HasDerivAt f (f' x) x ⊢ ∫ (x : ℝ) in b..a, f' x = f a - f b ** exact integral_eq_of_hasDerivWithinAt_off_countable_of_le f f' hab hs Hc Hd Hi.symm ** Qed
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MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff ** α : Type u_1 β : Type u_2 m : MeasurableSpace α c : ComplexMeasure α μ : VectorMeasure α ℝ≥0∞ ⊢ c ≪ᵥ μ ↔ ↑re c ≪ᵥ μ ∧ ↑im c ≪ᵥ μ ** constructor <;> intro h ** case mp.right α : Type u_1 β : Type u_2 m : MeasurableSpace α c : ComplexMeasure α μ : VectorMeasure α ℝ≥0∞ h : c ≪ᵥ μ ⊢ ↑im c ≪ᵥ μ ** intro i hi ** case mp.right α : Type u_1 β : Type u_2 m : MeasurableSpace α c : ComplexMeasure α μ : VectorMeasure α ℝ≥0∞ h : c ≪ᵥ μ i : Set α hi : ↑μ i = 0 ⊢ ↑(↑im c) i = 0 ** simp [h hi] ** case mpr α : Type u_1 β : Type u_2 m : MeasurableSpace α c : ComplexMeasure α μ : VectorMeasure α ℝ≥0∞ h : ↑re c ≪ᵥ μ ∧ ↑im c ≪ᵥ μ ⊢ c ≪ᵥ μ ** intro i hi ** case mpr α : Type u_1 β : Type u_2 m : MeasurableSpace α c : ComplexMeasure α μ : VectorMeasure α ℝ≥0∞ h : ↑re c ≪ᵥ μ ∧ ↑im c ≪ᵥ μ i : Set α hi : ↑μ i = 0 ⊢ ↑c i = 0 ** rw [← Complex.re_add_im (c i), (_ : (c i).re = 0), (_ : (c i).im = 0)] ** case mpr α : Type u_1 β : Type u_2 m : MeasurableSpace α c : ComplexMeasure α μ : VectorMeasure α ℝ≥0∞ h : ↑re c ≪ᵥ μ ∧ ↑im c ≪ᵥ μ i : Set α hi : ↑μ i = 0 ⊢ ↑0 + ↑0 * Complex.I = 0 α : Type u_1 β : Type u_2 m : MeasurableSpace α c : ComplexMeasure α μ : VectorMeasure α ℝ≥0∞ h : ↑re c ≪ᵥ μ ∧ ↑im c ≪ᵥ μ i : Set α hi : ↑μ i = 0 ⊢ (↑c i).im = 0 α : Type u_1 β : Type u_2 m : MeasurableSpace α c : ComplexMeasure α μ : VectorMeasure α ℝ≥0∞ h : ↑re c ≪ᵥ μ ∧ ↑im c ≪ᵥ μ i : Set α hi : ↑μ i = 0 ⊢ (↑c i).re = 0 ** exacts [by simp, h.2 hi, h.1 hi] ** α : Type u_1 β : Type u_2 m : MeasurableSpace α c : ComplexMeasure α μ : VectorMeasure α ℝ≥0∞ h : ↑re c ≪ᵥ μ ∧ ↑im c ≪ᵥ μ i : Set α hi : ↑μ i = 0 ⊢ ↑0 + ↑0 * Complex.I = 0 ** simp ** Qed
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StieltjesFunction.rightLim_eq ** f✝ f : StieltjesFunction x : ℝ ⊢ rightLim (↑f) x = ↑f x ** rw [← f.mono.continuousWithinAt_Ioi_iff_rightLim_eq, continuousWithinAt_Ioi_iff_Ici] ** f✝ f : StieltjesFunction x : ℝ ⊢ ContinuousWithinAt (↑f) (Ici x) x ** exact f.right_continuous' x ** Qed
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StieltjesFunction.iInf_Ioi_eq ** f✝ f : StieltjesFunction x : ℝ ⊢ ⨅ r, ↑f ↑r = ↑f x ** suffices Function.rightLim f x = ⨅ r : Ioi x, f r by rw [← this, f.rightLim_eq] ** f✝ f : StieltjesFunction x : ℝ ⊢ rightLim (↑f) x = ⨅ r, ↑f ↑r ** rw [f.mono.rightLim_eq_sInf, sInf_image'] ** f✝ f : StieltjesFunction x : ℝ ⊢ 𝓝[Ioi x] x ≠ ⊥ ** rw [← neBot_iff] ** f✝ f : StieltjesFunction x : ℝ ⊢ NeBot (𝓝[Ioi x] x) ** infer_instance ** f✝ f : StieltjesFunction x : ℝ this : rightLim (↑f) x = ⨅ r, ↑f ↑r ⊢ ⨅ r, ↑f ↑r = ↑f x ** rw [← this, f.rightLim_eq] ** Qed
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StieltjesFunction.countable_leftLim_ne ** f✝ f : StieltjesFunction ⊢ Set.Countable {x | leftLim (↑f) x ≠ ↑f x} ** refine Countable.mono ?_ f.mono.countable_not_continuousAt ** f✝ f : StieltjesFunction ⊢ {x | leftLim (↑f) x ≠ ↑f x} ⊆ {x | ¬ContinuousAt (↑f) x} ** intro x hx h'x ** f✝ f : StieltjesFunction x : ℝ hx : x ∈ {x | leftLim (↑f) x ≠ ↑f x} h'x : ContinuousAt (↑f) x ⊢ False ** apply hx ** f✝ f : StieltjesFunction x : ℝ hx : x ∈ {x | leftLim (↑f) x ≠ ↑f x} h'x : ContinuousAt (↑f) x ⊢ leftLim (↑f) x = ↑f x ** exact tendsto_nhds_unique (f.mono.tendsto_leftLim x) (h'x.tendsto.mono_left nhdsWithin_le_nhds) ** Qed
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StieltjesFunction.length_empty ** f : StieltjesFunction ⊢ ⨅ (_ : ∅ ⊆ Ioc 0 0), ofReal (↑f 0 - ↑f 0) ≤ 0 ** simp ** Qed
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StieltjesFunction.length_subadditive_Icc_Ioo ** f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) ⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ofReal (↑f (d i) - ↑f (c i)) ** suffices
∀ (s : Finset ℕ) (b), Icc a b ⊆ (⋃ i ∈ (s : Set ℕ), Ioo (c i) (d i)) →
(ofReal (f b - f a) : ℝ≥0∞) ≤ ∑ i in s, ofReal (f (d i) - f (c i)) by
rcases isCompact_Icc.elim_finite_subcover_image
(fun (i : ℕ) (_ : i ∈ univ) => @isOpen_Ioo _ _ _ _ (c i) (d i)) (by simpa using ss) with
⟨s, _, hf, hs⟩
have e : ⋃ i ∈ (hf.toFinset : Set ℕ), Ioo (c i) (d i) = ⋃ i ∈ s, Ioo (c i) (d i) := by
simp only [ext_iff, exists_prop, Finset.set_biUnion_coe, mem_iUnion, forall_const,
iff_self_iff, Finite.mem_toFinset]
rw [ENNReal.tsum_eq_iSup_sum]
refine' le_trans _ (le_iSup _ hf.toFinset)
exact this hf.toFinset _ (by simpa only [e] ) ** f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) ⊢ ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ** clear ss b ** f : StieltjesFunction a : ℝ c d : ℕ → ℝ ⊢ ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ** refine' fun s => Finset.strongInductionOn s fun s IH b cv => _ ** f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ cv : Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) ⊢ ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ** cases' le_total b a with ab ab ** case inr f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ cv : Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) ab : a ≤ b ⊢ ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ** have := cv ⟨ab, le_rfl⟩ ** case inr f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ cv : Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) ab : a ≤ b this : b ∈ ⋃ i ∈ ↑s, Ioo (c i) (d i) ⊢ ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ** simp only [Finset.mem_coe, gt_iff_lt, not_lt, ge_iff_le, mem_iUnion, mem_Ioo, exists_and_left,
exists_prop] at this ** case inr f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ cv : Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) ab : a ≤ b this : ∃ i, c i < b ∧ i ∈ s ∧ b < d i ⊢ ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ** rcases this with ⟨i, cb, is, bd⟩ ** case inr.intro.intro.intro f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ cv : Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) ab : a ≤ b i : ℕ cb : c i < b is : i ∈ s bd : b < d i ⊢ ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ** rw [← Finset.insert_erase is] at cv ⊢ ** case inr.intro.intro.intro f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ ab : a ≤ b i : ℕ cv : Icc a b ⊆ ⋃ i_1 ∈ ↑(insert i (Finset.erase s i)), Ioo (c i_1) (d i_1) cb : c i < b is : i ∈ s bd : b < d i ⊢ ofReal (↑f b - ↑f a) ≤ ∑ i in insert i (Finset.erase s i), ofReal (↑f (d i) - ↑f (c i)) ** rw [Finset.coe_insert, biUnion_insert] at cv ** case inr.intro.intro.intro f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ ab : a ≤ b i : ℕ cv : Icc a b ⊆ Ioo (c i) (d i) ∪ ⋃ x ∈ ↑(Finset.erase s i), Ioo (c x) (d x) cb : c i < b is : i ∈ s bd : b < d i ⊢ ofReal (↑f b - ↑f a) ≤ ∑ i in insert i (Finset.erase s i), ofReal (↑f (d i) - ↑f (c i)) ** rw [Finset.sum_insert (Finset.not_mem_erase _ _)] ** case inr.intro.intro.intro f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ ab : a ≤ b i : ℕ cv : Icc a b ⊆ Ioo (c i) (d i) ∪ ⋃ x ∈ ↑(Finset.erase s i), Ioo (c x) (d x) cb : c i < b is : i ∈ s bd : b < d i ⊢ ofReal (↑f b - ↑f a) ≤ ofReal (↑f (d i) - ↑f (c i)) + ∑ x in Finset.erase s i, ofReal (↑f (d x) - ↑f (c x)) ** refine' le_trans _ (add_le_add_left (IH _ (Finset.erase_ssubset is) (c i) _) _) ** f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) this : ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ofReal (↑f (d i) - ↑f (c i)) ** rcases isCompact_Icc.elim_finite_subcover_image
(fun (i : ℕ) (_ : i ∈ univ) => @isOpen_Ioo _ _ _ _ (c i) (d i)) (by simpa using ss) with
⟨s, _, hf, hs⟩ ** case intro.intro.intro f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) this : ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) s : Set ℕ left✝ : s ⊆ univ hf : Set.Finite s hs : Icc a b ⊆ ⋃ i ∈ s, Ioo (c i) (d i) ⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ofReal (↑f (d i) - ↑f (c i)) ** have e : ⋃ i ∈ (hf.toFinset : Set ℕ), Ioo (c i) (d i) = ⋃ i ∈ s, Ioo (c i) (d i) := by
simp only [ext_iff, exists_prop, Finset.set_biUnion_coe, mem_iUnion, forall_const,
iff_self_iff, Finite.mem_toFinset] ** case intro.intro.intro f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) this : ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) s : Set ℕ left✝ : s ⊆ univ hf : Set.Finite s hs : Icc a b ⊆ ⋃ i ∈ s, Ioo (c i) (d i) e : ⋃ i ∈ ↑(Finite.toFinset hf), Ioo (c i) (d i) = ⋃ i ∈ s, Ioo (c i) (d i) ⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ofReal (↑f (d i) - ↑f (c i)) ** rw [ENNReal.tsum_eq_iSup_sum] ** case intro.intro.intro f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) this : ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) s : Set ℕ left✝ : s ⊆ univ hf : Set.Finite s hs : Icc a b ⊆ ⋃ i ∈ s, Ioo (c i) (d i) e : ⋃ i ∈ ↑(Finite.toFinset hf), Ioo (c i) (d i) = ⋃ i ∈ s, Ioo (c i) (d i) ⊢ ofReal (↑f b - ↑f a) ≤ ⨆ s, ∑ a in s, ofReal (↑f (d a) - ↑f (c a)) ** refine' le_trans _ (le_iSup _ hf.toFinset) ** case intro.intro.intro f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) this : ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) s : Set ℕ left✝ : s ⊆ univ hf : Set.Finite s hs : Icc a b ⊆ ⋃ i ∈ s, Ioo (c i) (d i) e : ⋃ i ∈ ↑(Finite.toFinset hf), Ioo (c i) (d i) = ⋃ i ∈ s, Ioo (c i) (d i) ⊢ ofReal (↑f b - ↑f a) ≤ ∑ a in Finite.toFinset hf, ofReal (↑f (d a) - ↑f (c a)) ** exact this hf.toFinset _ (by simpa only [e] ) ** f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) this : ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ⊢ Icc ?m.97192 ?m.97193 ⊆ ⋃ i ∈ univ, Ioo (c i) (d i) ** simpa using ss ** f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) this : ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) s : Set ℕ left✝ : s ⊆ univ hf : Set.Finite s hs : Icc a b ⊆ ⋃ i ∈ s, Ioo (c i) (d i) ⊢ ⋃ i ∈ ↑(Finite.toFinset hf), Ioo (c i) (d i) = ⋃ i ∈ s, Ioo (c i) (d i) ** simp only [ext_iff, exists_prop, Finset.set_biUnion_coe, mem_iUnion, forall_const,
iff_self_iff, Finite.mem_toFinset] ** f : StieltjesFunction a b : ℝ c d : ℕ → ℝ ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i) this : ∀ (s : Finset ℕ) (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) s : Set ℕ left✝ : s ⊆ univ hf : Set.Finite s hs : Icc a b ⊆ ⋃ i ∈ s, Ioo (c i) (d i) e : ⋃ i ∈ ↑(Finite.toFinset hf), Ioo (c i) (d i) = ⋃ i ∈ s, Ioo (c i) (d i) ⊢ Icc a b ⊆ ⋃ i ∈ ↑(Finite.toFinset hf), Ioo (c i) (d i) ** simpa only [e] ** case inl f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ cv : Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) ab : b ≤ a ⊢ ofReal (↑f b - ↑f a) ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ** rw [ENNReal.ofReal_eq_zero.2 (sub_nonpos.2 (f.mono ab))] ** case inl f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ cv : Icc a b ⊆ ⋃ i ∈ ↑s, Ioo (c i) (d i) ab : b ≤ a ⊢ 0 ≤ ∑ i in s, ofReal (↑f (d i) - ↑f (c i)) ** exact zero_le _ ** case inr.intro.intro.intro.refine'_1 f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ ab : a ≤ b i : ℕ cv : Icc a b ⊆ Ioo (c i) (d i) ∪ ⋃ x ∈ ↑(Finset.erase s i), Ioo (c x) (d x) cb : c i < b is : i ∈ s bd : b < d i ⊢ ofReal (↑f b - ↑f a) ≤ ofReal (↑f (d i) - ↑f (c i)) + ofReal (↑f (c i) - ↑f a) ** refine' le_trans (ENNReal.ofReal_le_ofReal _) ENNReal.ofReal_add_le ** case inr.intro.intro.intro.refine'_1 f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ ab : a ≤ b i : ℕ cv : Icc a b ⊆ Ioo (c i) (d i) ∪ ⋃ x ∈ ↑(Finset.erase s i), Ioo (c x) (d x) cb : c i < b is : i ∈ s bd : b < d i ⊢ ↑f b - ↑f a ≤ ↑f (d i) - ↑f (c i) + (↑f (c i) - ↑f a) ** rw [sub_add_sub_cancel] ** case inr.intro.intro.intro.refine'_1 f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ ab : a ≤ b i : ℕ cv : Icc a b ⊆ Ioo (c i) (d i) ∪ ⋃ x ∈ ↑(Finset.erase s i), Ioo (c x) (d x) cb : c i < b is : i ∈ s bd : b < d i ⊢ ↑f b - ↑f a ≤ ↑f (d i) - ↑f a ** exact sub_le_sub_right (f.mono bd.le) _ ** case inr.intro.intro.intro.refine'_2 f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ ab : a ≤ b i : ℕ cv : Icc a b ⊆ Ioo (c i) (d i) ∪ ⋃ x ∈ ↑(Finset.erase s i), Ioo (c x) (d x) cb : c i < b is : i ∈ s bd : b < d i ⊢ Icc a (c i) ⊆ ⋃ i_1 ∈ ↑(Finset.erase s i), Ioo (c i_1) (d i_1) ** rintro x ⟨h₁, h₂⟩ ** case inr.intro.intro.intro.refine'_2.intro f : StieltjesFunction a : ℝ c d : ℕ → ℝ s✝ s : Finset ℕ IH : ∀ (t : Finset ℕ), t ⊂ s → ∀ (b : ℝ), Icc a b ⊆ ⋃ i ∈ ↑t, Ioo (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i in t, ofReal (↑f (d i) - ↑f (c i)) b : ℝ ab : a ≤ b i : ℕ cv : Icc a b ⊆ Ioo (c i) (d i) ∪ ⋃ x ∈ ↑(Finset.erase s i), Ioo (c x) (d x) cb : c i < b is : i ∈ s bd : b < d i x : ℝ h₁ : a ≤ x h₂ : x ≤ c i ⊢ x ∈ ⋃ i_1 ∈ ↑(Finset.erase s i), Ioo (c i_1) (d i_1) ** refine' (cv ⟨h₁, le_trans h₂ (le_of_lt cb)⟩).resolve_left (mt And.left (not_lt_of_le h₂)) ** Qed
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StieltjesFunction.measurableSet_Ioi ** f : StieltjesFunction c : ℝ ⊢ MeasurableSet (Ioi c) ** refine OuterMeasure.ofFunction_caratheodory fun t => ?_ ** f : StieltjesFunction c : ℝ t : Set ℝ ⊢ length f (t ∩ Ioi c) + length f (t \ Ioi c) ≤ length f t ** refine' le_iInf fun a => le_iInf fun b => le_iInf fun h => _ ** f : StieltjesFunction c : ℝ t : Set ℝ a b : ℝ h : t ⊆ Ioc a b ⊢ length f (t ∩ Ioi c) + length f (t \ Ioi c) ≤ ofReal (↑f b - ↑f a) ** refine'
le_trans
(add_le_add (f.length_mono <| inter_subset_inter_left _ h)
(f.length_mono <| diff_subset_diff_left h)) _ ** f : StieltjesFunction c : ℝ t : Set ℝ a b : ℝ h : t ⊆ Ioc a b ⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a) ** cases' le_total a c with hac hac <;> cases' le_total b c with hbc hbc ** case inl.inl f : StieltjesFunction c : ℝ t : Set ℝ a b : ℝ h : t ⊆ Ioc a b hac : a ≤ c hbc : b ≤ c ⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a) ** simp only [Ioc_inter_Ioi, f.length_Ioc, hac, _root_.sup_eq_max, hbc, le_refl, Ioc_eq_empty,
max_eq_right, min_eq_left, Ioc_diff_Ioi, f.length_empty, zero_add, not_lt] ** case inl.inr f : StieltjesFunction c : ℝ t : Set ℝ a b : ℝ h : t ⊆ Ioc a b hac : a ≤ c hbc : c ≤ b ⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a) ** simp only [hac, hbc, Ioc_inter_Ioi, Ioc_diff_Ioi, f.length_Ioc, min_eq_right,
_root_.sup_eq_max, ← ENNReal.ofReal_add, f.mono hac, f.mono hbc, sub_nonneg,
sub_add_sub_cancel, le_refl,
max_eq_right] ** case inr.inl f : StieltjesFunction c : ℝ t : Set ℝ a b : ℝ h : t ⊆ Ioc a b hac : c ≤ a hbc : b ≤ c ⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a) ** simp only [hbc, le_refl, Ioc_eq_empty, Ioc_inter_Ioi, min_eq_left, Ioc_diff_Ioi, f.length_empty,
zero_add, or_true_iff, le_sup_iff, f.length_Ioc, not_lt] ** case inr.inr f : StieltjesFunction c : ℝ t : Set ℝ a b : ℝ h : t ⊆ Ioc a b hac : c ≤ a hbc : c ≤ b ⊢ length f (Ioc a b ∩ Ioi c) + length f (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a) ** simp only [hac, hbc, Ioc_inter_Ioi, Ioc_diff_Ioi, f.length_Ioc, min_eq_right, _root_.sup_eq_max,
le_refl, Ioc_eq_empty, add_zero, max_eq_left, f.length_empty, not_lt] ** Qed
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StieltjesFunction.outer_trim ** f : StieltjesFunction ⊢ OuterMeasure.trim (StieltjesFunction.outer f) = StieltjesFunction.outer f ** refine' le_antisymm (fun s => _) (OuterMeasure.le_trim _) ** f : StieltjesFunction s : Set ℝ ⊢ ↑(OuterMeasure.trim (StieltjesFunction.outer f)) s ≤ ↑(StieltjesFunction.outer f) s ** rw [OuterMeasure.trim_eq_iInf] ** f : StieltjesFunction s : Set ℝ ⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t ≤ ↑(StieltjesFunction.outer f) s ** refine' le_iInf fun t => le_iInf fun ht => ENNReal.le_of_forall_pos_le_add fun ε ε0 h => _ ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t ≤ ∑' (i : ℕ), length f (t i) + ↑ε ** rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 ε0).ne' ℕ with ⟨ε', ε'0, hε⟩ ** case intro.intro f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε ⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t ≤ ∑' (i : ℕ), length f (t i) + ↑ε ** refine' le_trans _ (add_le_add_left (le_of_lt hε) _) ** case intro.intro f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε ⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t ≤ ∑' (i : ℕ), length f (t i) + ∑' (i : ℕ), ↑(ε' i) ** rw [← ENNReal.tsum_add] ** case intro.intro f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε ⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t ≤ ∑' (a : ℕ), (length f (t a) + ↑(ε' a)) ** choose g hg using
show ∀ i, ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal (ε' i) by
intro i
have hl :=
ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne'
conv at hl =>
lhs
rw [length]
simp only [iInf_lt_iff] at hl
rcases hl with ⟨a, b, h₁, h₂⟩
rw [← f.outer_Ioc] at h₂
exact ⟨_, h₁, measurableSet_Ioc, le_of_lt <| by simpa using h₂⟩ ** case intro.intro f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε g : ℕ → Set ℝ hg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ofReal ↑(ε' i) ⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t ≤ ∑' (a : ℕ), (length f (t a) + ↑(ε' a)) ** simp only [ofReal_coe_nnreal] at hg ** case intro.intro f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε g : ℕ → Set ℝ hg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i) ⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), ↑(StieltjesFunction.outer f) t ≤ ∑' (a : ℕ), (length f (t a) + ↑(ε' a)) ** apply iInf_le_of_le (iUnion g) _ ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε g : ℕ → Set ℝ hg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i) ⊢ ⨅ (_ : s ⊆ iUnion g), ⨅ (_ : MeasurableSet (iUnion g)), ↑(StieltjesFunction.outer f) (iUnion g) ≤ ∑' (a : ℕ), (length f (t a) + ↑(ε' a)) ** apply iInf_le_of_le (ht.trans <| iUnion_mono fun i => (hg i).1) _ ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε g : ℕ → Set ℝ hg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i) ⊢ ⨅ (_ : MeasurableSet (iUnion g)), ↑(StieltjesFunction.outer f) (iUnion g) ≤ ∑' (a : ℕ), (length f (t a) + ↑(ε' a)) ** apply iInf_le_of_le (MeasurableSet.iUnion fun i => (hg i).2.1) _ ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε g : ℕ → Set ℝ hg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ ↑(StieltjesFunction.outer f) (g i) ≤ length f (t i) + ↑(ε' i) ⊢ ↑(StieltjesFunction.outer f) (iUnion g) ≤ ∑' (a : ℕ), (length f (t a) + ↑(ε' a)) ** exact le_trans (f.outer.iUnion _) (ENNReal.tsum_le_tsum fun i => (hg i).2.2) ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε ⊢ ∀ (i : ℕ), ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i) ** intro i ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε i : ℕ ⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i) ** have hl :=
ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne' ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε i : ℕ hl : length f (t i) < length f (t i) + ↑(ε' i) ⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i) ** conv at hl =>
lhs
rw [length] ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε i : ℕ hl : ⨅ a, ⨅ b, ⨅ (_ : t i ⊆ Ioc a b), ofReal (↑f b - ↑f a) < length f (t i) + ↑(ε' i) ⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i) ** simp only [iInf_lt_iff] at hl ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε i : ℕ hl : ∃ i_1 i_2 i_3, ofReal (↑f i_2 - ↑f i_1) < length f (t i) + ↑(ε' i) ⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i) ** rcases hl with ⟨a, b, h₁, h₂⟩ ** case intro.intro.intro f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε i : ℕ a b : ℝ h₁ : t i ⊆ Ioc a b h₂ : ofReal (↑f b - ↑f a) < length f (t i) + ↑(ε' i) ⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i) ** rw [← f.outer_Ioc] at h₂ ** case intro.intro.intro f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε i : ℕ a b : ℝ h₁ : t i ⊆ Ioc a b h₂ : ↑(StieltjesFunction.outer f) (Ioc a b) < length f (t i) + ↑(ε' i) ⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ ↑(StieltjesFunction.outer f) s ≤ length f (t i) + ofReal ↑(ε' i) ** exact ⟨_, h₁, measurableSet_Ioc, le_of_lt <| by simpa using h₂⟩ ** f : StieltjesFunction s : Set ℝ t : ℕ → Set ℝ ht : s ⊆ ⋃ i, t i ε : ℝ≥0 ε0 : 0 < ε h : ∑' (i : ℕ), length f (t i) < ⊤ ε' : ℕ → ℝ≥0 ε'0 : ∀ (i : ℕ), 0 < ε' i hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε i : ℕ a b : ℝ h₁ : t i ⊆ Ioc a b h₂ : ↑(StieltjesFunction.outer f) (Ioc a b) < length f (t i) + ↑(ε' i) ⊢ ↑(StieltjesFunction.outer f) (Ioc a b) < length f (t i) + ofReal ↑(ε' i) ** simpa using h₂ ** Qed
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StieltjesFunction.borel_le_measurable ** f : StieltjesFunction ⊢ borel ℝ ≤ OuterMeasure.caratheodory (StieltjesFunction.outer f) ** rw [borel_eq_generateFrom_Ioi] ** f : StieltjesFunction ⊢ MeasurableSpace.generateFrom (range Ioi) ≤ OuterMeasure.caratheodory (StieltjesFunction.outer f) ** refine' MeasurableSpace.generateFrom_le _ ** f : StieltjesFunction ⊢ ∀ (t : Set ℝ), t ∈ range Ioi → MeasurableSet t ** simp (config := { contextual := true }) [f.measurableSet_Ioi] ** Qed
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StieltjesFunction.measure_Ioc ** f : StieltjesFunction a b : ℝ ⊢ ↑↑(StieltjesFunction.measure f) (Ioc a b) = ofReal (↑f b - ↑f a) ** rw [StieltjesFunction.measure] ** f : StieltjesFunction a b : ℝ ⊢ ↑↑{ toOuterMeasure := StieltjesFunction.outer f, m_iUnion := (_ : ∀ (_s : ℕ → Set ℝ), (∀ (i : ℕ), MeasurableSet (_s i)) → Pairwise (Disjoint on _s) → ↑(StieltjesFunction.outer f) (⋃ i, _s i) = ∑' (i : ℕ), ↑(StieltjesFunction.outer f) (_s i)), trimmed := (_ : OuterMeasure.trim (StieltjesFunction.outer f) = StieltjesFunction.outer f) } (Ioc a b) = ofReal (↑f b - ↑f a) ** exact f.outer_Ioc a b ** Qed
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