diff --git a/Analysis/MeasureTheory/Section_1_2_2.lean b/Analysis/MeasureTheory/Section_1_2_2.lean index ebca1f47..9f66bc42 100644 --- a/Analysis/MeasureTheory/Section_1_2_2.lean +++ b/Analysis/MeasureTheory/Section_1_2_2.lean @@ -1657,23 +1657,18 @@ example {d:ℕ} (m: Set (EuclideanSpace' d) → EReal) (h_empty: m ∅ = 0) (h_p sorry /-- Exercise 1.2.13(i) -/ -example {d:ℕ} {E: ℕ → Set (EuclideanSpace' d)} (hE: ∀ n, LebesgueMeasurable (E n)) : - ∃ E₀, LebesgueMeasurable E₀ ∧ - IsNull {x | ¬ Filter.atTop.Tendsto (fun n ↦ (E n).indicator' x) (nhds (E₀.indicator' x))} := by sorry +example {d:ℕ} {E: ℕ → Set (EuclideanSpace' d)} {E₀: Set (EuclideanSpace' d)} (hE: ∀ n, LebesgueMeasurable (E n)) (hpoint: ∀ x, Filter.atTop.Tendsto (fun n ↦ (E n).indicator' x) (nhds (E₀.indicator' x))) : LebesgueMeasurable E₀ := by sorry /-- Exercise 1.2.13(ii) -/ -example {d:ℕ} {E: ℕ → Set (EuclideanSpace' d)} {F: Set (EuclideanSpace' d)} +example {d:ℕ} {E: ℕ → Set (EuclideanSpace' d)} {E₀ F: Set (EuclideanSpace' d)} (hE: ∀ n, LebesgueMeasurable (E n)) - (hsub: ∀ n, E n ⊆ F) (hFmes: LebesgueMeasurable F) (hfin: Lebesgue_measure F < ⊤) : - ∃ E₀, LebesgueMeasurable E₀ ∧ - IsNull {x | ¬ Filter.atTop.Tendsto (fun n ↦ (E n).indicator' x) (nhds (E₀.indicator' x))} ∧ - Filter.atTop.Tendsto (fun n ↦ Lebesgue_measure (E n)) (nhds (Lebesgue_measure E₀)) := by sorry + (hpoint: ∀ x, Filter.atTop.Tendsto (fun n ↦ (E n).indicator' x) (nhds (E₀.indicator' x))) + (hsub: ∀ n, E n ⊆ F) (hFmes: LebesgueMeasurable F) (hfin: Lebesgue_measure F < ⊤) : Filter.atTop.Tendsto (fun n ↦ Lebesgue_measure (E n)) (nhds (Lebesgue_measure E₀)) := by sorry /-- Exercise 1.2.13(iii) -/ example : ∃ (d:ℕ) (E: ℕ → Set (EuclideanSpace' d)) (E₀ F: Set (EuclideanSpace' d)) (hE: ∀ n, LebesgueMeasurable (E n)) - (hE₀: LebesgueMeasurable E₀) - (hpoint : IsNull {x | ¬ Filter.atTop.Tendsto (fun n ↦ (E n).indicator' x) (nhds (E₀.indicator' x))}) + (hpoint: ∀ x, Filter.atTop.Tendsto (fun n ↦ (E n).indicator' x) (nhds (E₀.indicator' x))) (hsub: ∀ n, E n ⊆ F) (hFmes: LebesgueMeasurable F), ¬ Filter.atTop.Tendsto (fun n ↦ Lebesgue_measure (E n)) (nhds (Lebesgue_measure E₀)) := by sorry /-- Exercise 1.2.14 -/