| import OAI.Probability.FreeIsing.Estimates.InfinitePiPair | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory MeasureTheory.Measure Set | |
| namespace Problem367 | |
| variable {I : Type*} {A B : I → Type*} | |
| [∀ i, MeasurableSpace (A i)] [∀ i, MeasurableSpace (B i)] | |
| def dependentPiPairEquiv : (∀ i, A i × B i) ≃ᵐ ((∀ i, A i) × (∀ i, B i)) where | |
| toFun x := (fun i => (x i).1, fun i => (x i).2) | |
| invFun p := fun i => (p.1 i, p.2 i) | |
| left_inv _ := rfl | |
| right_inv _ := rfl | |
| measurable_toFun := by | |
| change Measurable (fun x : ∀ i, A i × B i => | |
| (fun i => (x i).1, fun i => (x i).2)) | |
| fun_prop | |
| measurable_invFun := by | |
| change Measurable (fun p : (∀ i, A i) × (∀ i, B i) => | |
| fun i => (p.1 i, p.2 i)) | |
| fun_prop | |
| theorem measurePreserving_dependentPiPairEquiv [Fintype I] | |
| (μ : ∀ i, Measure (A i)) (ν : ∀ i, Measure (B i)) | |
| [∀ i, SigmaFinite (μ i)] [∀ i, SigmaFinite (ν i)] : | |
| MeasurePreserving (dependentPiPairEquiv (A := A) (B := B)) | |
| (Measure.pi fun i => (μ i).prod (ν i)) | |
| ((Measure.pi μ).prod (Measure.pi ν)) where | |
| measurable := dependentPiPairEquiv.measurable | |
| map_eq := by | |
| refine (FiniteSpanningSetsIn.ext ?_ (isPiSystem_pi.prod isPiSystem_pi) | |
| ((FiniteSpanningSetsIn.pi fun i => (μ i).toFiniteSpanningSetsIn).prod | |
| (FiniteSpanningSetsIn.pi fun i => (ν i).toFiniteSpanningSetsIn)) ?_).symm | |
| · refine (generateFrom_eq_prod generateFrom_pi generateFrom_pi ?_ ?_).symm | |
| · exact (FiniteSpanningSetsIn.pi fun i => | |
| (μ i).toFiniteSpanningSetsIn).isCountablySpanning | |
| · exact (FiniteSpanningSetsIn.pi fun i => | |
| (ν i).toFiniteSpanningSetsIn).isCountablySpanning | |
| · rintro _ ⟨s, ⟨s, _, rfl⟩, ⟨_, ⟨t, _, rfl⟩, rfl⟩⟩ | |
| rw [MeasurableEquiv.map_apply] | |
| have heq : dependentPiPairEquiv ⁻¹' (univ.pi s ×ˢ univ.pi t) = | |
| univ.pi (fun i => s i ×ˢ t i) := by | |
| ext x | |
| simp [dependentPiPairEquiv, mem_pi, forall_and] | |
| rw [heq] | |
| simp_rw [Measure.pi_pi, Measure.prod_prod, Measure.pi_pi, Finset.prod_mul_distrib] | |
| theorem infinitePi_map_pair_dependent_arrays | |
| (μ : ∀ i, Measure (A i)) (ν : ∀ i, Measure (B i)) | |
| [∀ i, IsProbabilityMeasure (μ i)] [∀ i, IsProbabilityMeasure (ν i)] : | |
| ((Measure.infinitePi μ).prod (Measure.infinitePi ν)).map | |
| (fun p : (∀ i, A i) × (∀ i, B i) => fun i => (p.1 i, p.2 i)) = | |
| Measure.infinitePi (fun i => (μ i).prod (ν i)) := by | |
| apply (Measure.isProjectiveLimit_infinitePi (fun i => (μ i).prod (ν i))).unique ?_ |>.symm | |
| intro s | |
| rw [Measure.map_map (by fun_prop) (by fun_prop)] | |
| have hcomp : | |
| s.restrict ∘ (fun p : (∀ i, A i) × (∀ i, B i) => fun i => (p.1 i, p.2 i)) = | |
| (fun p : (∀ i : s, A i) × (∀ i : s, B i) => fun i => (p.1 i, p.2 i)) ∘ | |
| Prod.map s.restrict s.restrict := rfl | |
| rw [hcomp, ← Measure.map_map (by fun_prop) (by fun_prop), | |
| ← Measure.map_prod_map _ _ (by fun_prop) (by fun_prop), | |
| Measure.infinitePi_map_restrict, Measure.infinitePi_map_restrict] | |
| exact (measurePreserving_dependentPiPairEquiv | |
| (fun i : s => μ i) (fun i : s => ν i)).symm.map_eq | |
| theorem infinitePi_map_split_dependent_pairs | |
| (μ : ∀ i, Measure (A i)) (ν : ∀ i, Measure (B i)) | |
| [∀ i, IsProbabilityMeasure (μ i)] [∀ i, IsProbabilityMeasure (ν i)] : | |
| (Measure.infinitePi (fun i => (μ i).prod (ν i))).map | |
| (fun x : ∀ i, A i × B i => (fun i => (x i).1, fun i => (x i).2)) = | |
| (Measure.infinitePi μ).prod (Measure.infinitePi ν) := by | |
| rw [← infinitePi_map_pair_dependent_arrays μ ν, | |
| Measure.map_map (by fun_prop) (by fun_prop)] | |
| have hcomp : | |
| (fun x : ∀ i, A i × B i => (fun i => (x i).1, fun i => (x i).2)) ∘ | |
| (fun p : (∀ i, A i) × (∀ i, B i) => fun i => (p.1 i, p.2 i)) = id := rfl | |
| rw [hcomp, Measure.map_id] | |
| end Problem367 | |
| end | |
| end OAI | |