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fix: Ex. 1.2.18/1.2.24 duplicate doc labels #574

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fix: Ex. 1.2.18/1.2.24 duplicate doc labels #574

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515c95d
fix: deduplicate Ex. 1.2.18/1.2.24 doc labels
Chessing234 Jul 8, 2026
989399d
fix: split Ex. 1.2.12 and 1.2.22 doc labels
Chessing234 Jul 8, 2026
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14 changes: 6 additions & 8 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 -/
/-- 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

Expand Down Expand Up @@ -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
Expand Down Expand Up @@ -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
Expand All @@ -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
Expand All @@ -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

Expand Down
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