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Fill that measurable indicators are simple functions with the expected integral #717

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Fill that measurable indicators are simple functions with the expected integral #717

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ddec488
Record that the EReal indicator is toEReal after the real indicator.
Chessing234 Sep 8, 2026
a8fdc7f
Keep the off-set evaluation of the EReal indicator.
Chessing234 Sep 8, 2026
15f4027
Fill that measurable indicators are unsigned simple functions.
Chessing234 Sep 8, 2026
9ddebf6
Fill that the unsigned simple integral of an indicator is Lebesgue me…
Chessing234 Sep 8, 2026
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38 changes: 34 additions & 4 deletions Analysis/MeasureTheory/Section_1_3_1.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -27,6 +27,8 @@ theorem EReal.indicator_of_mem {X:Type*} {A: Set X} {x:X} (h: x ∈ A) : EReal.i
theorem EReal.indicator_of_notMem {X:Type*} {A: Set X} {x:X} (h: x ∉ A) : EReal.indicator A x = 0 := by
simp [EReal.indicator, Real.EReal_fun, Set.indicator'_of_notMem h]

theorem EReal.indicator_eq_comp {X:Type*} (A: Set X) : EReal.indicator A = Real.toEReal ∘ A.indicator' := rfl

noncomputable def Complex.indicator {X:Type*} (A: Set X) : X → ℂ := Real.complex_fun A.indicator'

/-- Definition 1.3.2 -/
Expand Down Expand Up @@ -1062,12 +1064,26 @@ lemma UnsignedSimpleFunction.integral_le_integral_of_aeLe {d:ℕ} {f g: Euclidea
/-- Exercise 1.3.1(vi) (Compatibility with Lebesgue measure, indicator) -/
lemma UnsignedSimpleFunction.indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E) :
UnsignedSimpleFunction (Real.toEReal ∘ E.indicator') := by
sorry
use 1, fun _ => (1 : EReal), fun _ => E
constructor
· intro
exact ⟨hE, zero_le_one⟩
· ext x
simp only [Function.comp_apply, Finset.sum_apply, Pi.smul_apply, smul_eq_mul]
rw [Fin.sum_univ_one]
simp [EReal.indicator, Real.EReal_fun]

/-- Exercise 1.3.1(vi) (Compatibility with Lebesgue measure, integral of an indicator) -/
lemma UnsignedSimpleFunction.integral_indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E) :
(UnsignedSimpleFunction.indicator hE).integ = Lebesgue_measure E := by
sorry
have hrep : Real.toEReal ∘ E.indicator' =
∑ i : Fin 1, (1 : EReal) • EReal.indicator ((fun _ : Fin 1 => E) i) := by
ext x
simp only [Function.comp_apply, Finset.sum_apply, Pi.smul_apply, smul_eq_mul]
rw [Fin.sum_univ_one]
simp [EReal.indicator, Real.EReal_fun]
have h := integral_eq (indicator hE) (fun _ => hE) (fun _ => zero_le_one) hrep
simpa [Fin.sum_univ_one] using h

lemma RealSimpleFunction.abs {d:ℕ} {f: EuclideanSpace' d → ℝ} (hf: RealSimpleFunction f) : UnsignedSimpleFunction (EReal.abs_fun f) := by
sorry
Expand Down Expand Up @@ -1299,11 +1315,25 @@ lemma ComplexSimpleFunction.integral_eq_integral_of_aeEqual {d:ℕ} {f g: Euclid
/-- Exercise 1.3.2(iii) (Compatibility with Lebesgue measure, indicator) -/
lemma RealSimpleFunction.indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E) :
RealSimpleFunction (E.indicator') := by
sorry
use 1, fun _ => (1 : ℝ), fun _ => E
constructor
· intro
exact hE
· ext x
simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul]
rw [Fin.sum_univ_one]
simp

lemma ComplexSimpleFunction.indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E) :
ComplexSimpleFunction (Complex.indicator E) := by
sorry
use 1, fun _ => (1 : ℂ), fun _ => E
constructor
· intro
exact hE
· ext x
simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul]
rw [Fin.sum_univ_one]
simp [Complex.indicator, Real.complex_fun]

/-- Exercise 1.3.2(iii) (Compatibility with Lebesgue measure, integral of an indicator) -/
lemma RealSimpleFunction.integral_indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E) (hfin: Lebesgue_measure E < ⊤): (RealSimpleFunction.indicator hE).integ = (Lebesgue_measure E).toReal := by
Expand Down
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