| import OAI.Analysis.IrrationalRotation.Analysis.FiniteUniformLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| open scoped Classical |
|
|
| namespace Invariant233.FiniteUniformLaw |
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|
| 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 |
|
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| end |
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| end OAI |
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|
