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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 | import OAI.Analysis.IrrationalRotation.Analysis.FiniteUniformLaw
namespace OAI
noncomputable section
open MeasureTheory
open scoped Classical
namespace Invariant233.FiniteUniformLaw
theorem infinitePi_fin_uniform (α : ℕ → Type*)
[∀ i, Fintype (α i)] [∀ i, Nonempty (α i)]
[∀ i, MeasurableSpace (α i)] [∀ i, MeasurableSingletonClass (α i)] (N : ℕ) :
(Measure.infinitePi (fun i => (PMF.uniformOfFintype (α i)).toMeasure)).map
(fun d (i : Fin N) => d i) =
(PMF.uniformOfFintype ((i : Fin N) → α i)).toMeasure := by
classical
let e : (Finset.range N) ≃ Fin N :=
{ toFun := fun i => ⟨i.1, Finset.mem_range.mp i.2⟩
invFun := fun i => ⟨i.1, Finset.mem_range.mpr i.2⟩
left_inv := fun _ => rfl
right_inv := fun _ => rfl }
let E : ((i : Finset.range N) → α i) ≃ ((i : Fin N) → α i) :=
Equiv.piCongrLeft (fun i : Fin N => α i) e
have hcomp : (fun d (i : Fin N) => d i) =
E ∘ (Finset.range N).restrict := by rfl
rw [hcomp, ← Measure.map_map (measurable_of_finite E) (Finset.measurable_restrict _),
infinitePi_restrict_uniform]
apply Measure.ext_of_singleton
intro a
rw [Measure.map_apply (measurable_of_finite E) (measurableSet_singleton a)]
have he : E ⁻¹' {a} = {E.symm a} := by
ext b
exact E.eq_symm_apply.symm
rw [he, PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton _),
PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton _)]
simp only [PMF.uniformOfFintype_apply]
congr 2
simp only [← Nat.card_eq_fintype_card]
exact Nat.card_congr E
alias infinitePi_fin_prefix_uniform := infinitePi_fin_uniform
end Invariant233.FiniteUniformLaw
end
end OAI
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