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fix: Exercise 1.1.7(i) needs a closed box #648

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fix: Exercise 1.1.7(i) needs a closed box #648

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13 changes: 10 additions & 3 deletions Analysis/MeasureTheory/Section_1_1_2.lean
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Expand Up @@ -660,15 +660,22 @@ lemma JordanMeasurable.measure_of_translate {d:ℕ} {E: Set (EuclideanSpace' d)}
· exact eq_outer hE;

/-- Exercise 1.1.7 (i) (Regions under graphs are Jordan measurable) -/
lemma JordanMeasurable.graph {d:ℕ} {B:Box d} {f: EuclideanSpace' d → ℝ} (hf: ContinuousOn f B.toSet) : JordanMeasurable { p | ∃ x ∈ B.toSet, EuclideanSpace'.prod_equiv d 1 p = ⟨ x, f x ⟩ } := by
lemma JordanMeasurable.graph {d:ℕ} {B:Box d} (hB: IsClosed B.toSet) {f: EuclideanSpace' d → ℝ}
(hf: ContinuousOn f B.toSet) :
JordanMeasurable { p | ∃ x ∈ B.toSet, EuclideanSpace'.prod_equiv d 1 p = ⟨ x, f x ⟩ } := by
sorry

/-- Exercise 1.1.7 (i) (Regions under graphs are Jordan measurable) -/
lemma JordanMeasurable.measure_of_graph {d:ℕ} {B:Box d} {f: EuclideanSpace' d → ℝ} (hf: ContinuousOn f B.toSet) : (JordanMeasurable.graph hf).measure = 0 := by
lemma JordanMeasurable.measure_of_graph {d:ℕ} {B:Box d} (hB: IsClosed B.toSet)
{f: EuclideanSpace' d → ℝ} (hf: ContinuousOn f B.toSet) :
(JordanMeasurable.graph hB hf).measure = 0 := by
sorry

/-- Exercise 1.1.7 (i) (Regions under graphs are Jordan measurable) -/
lemma JordanMeasurable.undergraph {d:ℕ} {B:Box d} {f: EuclideanSpace' d → ℝ} (hf: ContinuousOn f B.toSet) : JordanMeasurable { p | ∃ x ∈ B.toSet, ∃ t:ℝ, EuclideanSpace'.prod_equiv d 1 p = ⟨ x, t ⟩ ∧ 0 ≤ t ∧ t ≤ f x } := by
lemma JordanMeasurable.undergraph {d:ℕ} {B:Box d} (hB: IsClosed B.toSet)
{f: EuclideanSpace' d → ℝ} (hf: ContinuousOn f B.toSet) :
JordanMeasurable
{ p | ∃ x ∈ B.toSet, ∃ t:ℝ, EuclideanSpace'.prod_equiv d 1 p = ⟨ x, t ⟩ ∧ 0 ≤ t ∧ t ≤ f x } := by
sorry

/-- Exercise 1.1.8(i) (A triangle is Jordan measurable) -/
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