diff --git a/Analysis/MeasureTheory/Section_1_2_2.lean b/Analysis/MeasureTheory/Section_1_2_2.lean index 9247f5b4..8a8ca3da 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(i) (Monotonicity)-/ +/-- 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(ii) (σ-subadditivity)-/ +/-- 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 @@ -1754,11 +1754,11 @@ 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(i) (Outer measure product bound)-/ +/-- 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(ii) (Measurability of product)-/ +/-- 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