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fix: space Verso docstring closers in Section 1.2.2 #621

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fix: space Verso docstring closers in Section 1.2.2 #621

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b0e08e5
fix: space Verso closers on Ex. 1.2.12(i)/(ii)
Chessing234 Aug 3, 2026
570f415
fix: space Verso closers on Ex. 1.2.12/1.2.22 docstrings
Chessing234 Aug 3, 2026
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8 changes: 4 additions & 4 deletions Analysis/MeasureTheory/Section_1_2_2.lean
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Expand Up @@ -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

Expand Down Expand Up @@ -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

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