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f61e547 | 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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | 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
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