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[Merged by Bors] - refactor(MeasureTheory): golf Mathlib/MeasureTheory/Constructions/Pi #38353

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[Merged by Bors] - refactor(MeasureTheory): golf Mathlib/MeasureTheory/Constructions/Pi #38353

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refactor(MeasureTheory): golf Pi construction
yuanyi-350 Apr 22, 2026
35ff69a
fix ci
yuanyi-350 Apr 22, 2026
0b21b67
golf
yuanyi-350 Apr 24, 2026
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23 changes: 6 additions & 17 deletions Mathlib/MeasureTheory/Constructions/Pi.lean
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Expand Up @@ -450,23 +450,6 @@ theorem ae_eq_set_pi {I : Set ι} {s t : ∀ i, Set (α i)} (h : ∀ i ∈ I, s
Set.pi I s =ᵐ[Measure.pi μ] Set.pi I t :=
(ae_le_set_pi fun i hi => (h i hi).le).antisymm (ae_le_set_pi fun i hi => (h i hi).symm.le)

lemma pi_map_piCongrLeft [hι' : Fintype ι'] (e : ι ≃ ι') {β : ι' → Type*}
[∀ i, MeasurableSpace (β i)] (μ : (i : ι') → Measure (β i)) [∀ i, SigmaFinite (μ i)] :
(Measure.pi fun i ↦ μ (e i)).map (MeasurableEquiv.piCongrLeft (fun i ↦ β i) e)
= Measure.pi μ := by
let e_meas : ((b : ι) → β (e b)) ≃ᵐ ((a : ι') → β a) :=
MeasurableEquiv.piCongrLeft (fun i ↦ β i) e
refine Measure.pi_eq (fun s _ ↦ ?_) |>.symm
rw [e_meas.measurableEmbedding.map_apply]
let s' : (i : ι) → Set (β (e i)) := fun i ↦ s (e i)
have : e_meas ⁻¹' pi univ s = pi univ s' := by
ext x
simp only [mem_preimage, Set.mem_pi, mem_univ, forall_true_left, s']
refine (e.forall_congr ?_).symm
intro i
rw [MeasurableEquiv.piCongrLeft_apply_apply e x i]
simpa [this] using Fintype.prod_equiv _ (fun _ ↦ (μ _) (s' _)) _ (congrFun rfl)

lemma pi_map_piOptionEquivProd {β : Option ι → Type*} [∀ i, MeasurableSpace (β i)]
(μ : (i : Option ι) → Measure (β i)) [∀ (i : Option ι), SigmaFinite (μ i)] :
((Measure.pi fun i ↦ μ (some i)).prod (μ none)).map
Expand Down Expand Up @@ -755,6 +738,12 @@ theorem volume_measurePreserving_piCongrLeft (α : ι → Type*) (f : ι' ≃ ι
MeasurePreserving (MeasurableEquiv.piCongrLeft α f) volume volume :=
measurePreserving_piCongrLeft (fun _ ↦ volume) f

lemma Measure.pi_map_piCongrLeft (e : ι ≃ ι') {β : ι' → Type*} [∀ i, MeasurableSpace (β i)]
(μ : (i : ι') → Measure (β i)) [∀ i, SigmaFinite (μ i)] :
(Measure.pi fun i ↦ μ (e i)).map (MeasurableEquiv.piCongrLeft (fun i ↦ β i) e) =
Measure.pi μ :=
(measurePreserving_piCongrLeft (α := fun i ↦ β i) μ e).map_eq

theorem measurePreserving_arrowProdEquivProdArrow (α β γ : Type*) [MeasurableSpace α]
[MeasurableSpace β] [Fintype γ] (μ : γ → Measure α) (ν : γ → Measure β) [∀ i, SigmaFinite (μ i)]
[∀ i, SigmaFinite (ν i)] :
Expand Down
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