diff --git a/Analysis/MeasureTheory/Section_1_2_2.lean b/Analysis/MeasureTheory/Section_1_2_2.lean index fee9dfe6..6376d9b9 100644 --- a/Analysis/MeasureTheory/Section_1_2_2.lean +++ b/Analysis/MeasureTheory/Section_1_2_2.lean @@ -1647,12 +1647,12 @@ theorem Lebesgue_measure.downward_monotone_convergence {d:ℕ} {E: ℕ → Set ( /-- Exercise 1.2.11 (c) (counterexample)-/ example : ∃ (d:ℕ) (E: ℕ → Set (EuclideanSpace' d)) (hE: ∀ n, LebesgueMeasurable (E n)) (hmono: ∀ n, E (n+1) ⊆ E n), ¬ Filter.atTop.Tendsto (fun n ↦ Lebesgue_measure (E n)) (nhds (Lebesgue_measure (⋂ n, E n))) := by sorry -/-- Exercise 1.2.12 -/ +/-- Exercise 1.2.12(i) (Monotonicity)-/ example {d:ℕ} (m: Set (EuclideanSpace' d) → EReal) (h_empty: m ∅ = 0) (h_pos: ∀ E, 0 ≤ m E) (hadd: ∀ E: ℕ → Set (EuclideanSpace' d), (Set.univ.PairwiseDisjoint E) → (∀ n, LebesgueMeasurable (E n)) → m (⋃ n, E n) = ∑' n, m (E n)) {E F: Set (EuclideanSpace' d)} (hsub: E ⊆ F) (hE: LebesgueMeasurable E) (hF: LebesgueMeasurable F) : m E ≤ m F := by sorry -/-- Exercise 1.2.12 -/ +/-- Exercise 1.2.12(ii) (σ-subadditivity)-/ example {d:ℕ} (m: Set (EuclideanSpace' d) → EReal) (h_empty: m ∅ = 0) (h_pos: ∀ E, 0 ≤ m E) (hadd: ∀ E: ℕ → Set (EuclideanSpace' d), (Set.univ.PairwiseDisjoint E) → (∀ n, LebesgueMeasurable (E n)) → m (⋃ n, E n) = ∑' n, m (E n)) {E: ℕ → Set (EuclideanSpace' d)} (hE: ∀ n, LebesgueMeasurable (E n)): m (⋃ n, E n) ≤ ∑' n, m (E n) := by sorry @@ -1716,7 +1716,7 @@ theorem inner_measure.le {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: Bornology.IsB : inner_measure hE ≤ Lebesgue_outer_measure E := by sorry -/-- Exercise 1.2.18(ii) (Inner measure)-/ +/-- Exercise 1.2.18(iii) (Inner measure)-/ theorem inner_measure.eq_iff {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: Bornology.IsBounded E) : inner_measure hE = Lebesgue_outer_measure E ↔ LebesgueMeasurable E := by sorry @@ -1754,15 +1754,15 @@ lemma Lebesgue_measure.linear {d:ℕ} (A: Matrix (Fin d) (Fin d) ℝ) [Invertibl {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E): Lebesgue_measure (A.linear_equiv '' E) = |A.det| * Lebesgue_measure E := by sorry -/-- Exercise 1.2.22 -/ +/-- Exercise 1.2.22(i) (Outer measure product bound)-/ theorem Lebesgue_outer_measure.prod {d₁ d₂:ℕ} {E₁: Set (EuclideanSpace' d₁)} {E₂: Set (EuclideanSpace' d₂)} : Lebesgue_outer_measure (EuclideanSpace'.prod E₁ E₂) ≤ Lebesgue_outer_measure E₁ * Lebesgue_outer_measure E₂ := by sorry -/-- Exercise 1.2.22 -/ +/-- Exercise 1.2.22(ii) (Measurability of product)-/ theorem LebesgueMeasurable.prod {d₁ d₂:ℕ} {E₁: Set (EuclideanSpace' d₁)} {E₂: Set (EuclideanSpace' d₂)} (hE₁: LebesgueMeasurable E₁) (hE₂: LebesgueMeasurable E₂) : LebesgueMeasurable (EuclideanSpace'.prod E₁ E₂) := by sorry -/-- Exercise 1.2.22 -/ +/-- Exercise 1.2.22(iii) (Product measure formula)-/ theorem Lebesgue_measure.prod {d₁ d₂:ℕ} {E₁: Set (EuclideanSpace' d₁)} {E₂: Set (EuclideanSpace' d₂)} (hE₁: LebesgueMeasurable E₁) (hE₂: LebesgueMeasurable E₂) : Lebesgue_measure (EuclideanSpace'.prod E₁ E₂) = Lebesgue_measure E₁ * Lebesgue_measure E₂ := by sorry @@ -1789,7 +1789,6 @@ def IsElementary.ae_quot {d:ℕ} {A: Set (EuclideanSpace' d)} (hA: IsElementary /-- Exercise 1.2.24(ii) (Lebesgue measure as the completion of elementary measure)-/ noncomputable def IsElementary.dist {d:ℕ} {A: Set (EuclideanSpace' d)} (hA: IsElementary A) : hA.ae_subsets → hA.ae_subsets → ℝ := Quotient.lift₂ (fun E F ↦ (Lebesgue_outer_measure (Subtype.val '' (_root_.symmDiff E F))).toReal) (by sorry) -/-- Exercise 1.2.24(ii) (Lebesgue measure as the completion of elementary measure)-/ noncomputable instance IsElementary.metric {d:ℕ} {A: Set (EuclideanSpace' d)} (hA: IsElementary A) : MetricSpace hA.ae_subsets := { dist := hA.dist dist_self := by sorry @@ -1798,7 +1797,6 @@ noncomputable instance IsElementary.metric {d:ℕ} {A: Set (EuclideanSpace' d)} dist_triangle := by sorry } -/-- Exercise 1.2.24(ii) (Lebesgue measure as the completion of elementary measure)-/ instance IsElementary.complete {d:ℕ} {A: Set (EuclideanSpace' d)} (hA: IsElementary A) : CompleteSpace hA.ae_subsets := by sorry