File size: 3,137 Bytes
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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ProbabilityTheory
namespace Problem367
variable {W X Y : Type*} [MeasurableSpace W] [MeasurableSpace X]
[MeasurableSpace Y]
lemma map_compProd_prodMap (mu : Measure W) [SFinite mu]
(kappa : Kernel W X) [IsSFiniteKernel kappa]
(T : W ≃ᵐ W) {S : X → Y} (hS : Measurable S) :
(mu ⊗ₘ kappa).map (Prod.map T S) =
mu.map T ⊗ₘ (kappa.comap T.symm T.symm.measurable).map S := by
ext A hA
rw [Measure.map_apply (T.measurable.prodMap hS) hA,
Measure.compProd_apply ((T.measurable.prodMap hS) hA),
Measure.compProd_apply hA,
lintegral_map (Kernel.measurable_kernel_prodMk_left hA) T.measurable]
apply lintegral_congr
intro w
rw [Kernel.map_apply' _ hS _ (measurable_prodMk_left hA), Kernel.comap_apply]
simp only [MeasurableEquiv.symm_apply_apply]
rfl
lemma kernel_ae_equivariant_of_joint_invariant
(mu : Measure W) [IsFiniteMeasure mu]
(kappa : Kernel W X) [IsFiniteKernel kappa]
[MeasurableSpace.CountableOrCountablyGenerated W X]
(T : W ≃ᵐ W) {S : X → X} (hS : Measurable S)
(hT : MeasurePreserving T mu mu)
(hjoint : (mu ⊗ₘ kappa).map (Prod.map T S) = mu ⊗ₘ kappa) :
∀ᵐ w ∂mu, kappa (T w) = (kappa w).map S := by
have heq : (kappa.comap T.symm T.symm.measurable).map S =ᵐ[mu] kappa := by
apply Kernel.ae_eq_of_compProd_eq
have h := hjoint
rw [map_compProd_prodMap mu kappa T hS, hT.map_eq] at h
exact h
have htrans := hT.quasiMeasurePreserving.ae heq
filter_upwards [htrans] with w hw
simpa only [Kernel.map_apply _ hS, Kernel.comap_apply,
MeasurableEquiv.symm_apply_apply] using hw.symm
lemma joint_invariant_of_kernel_ae_equivariant
(mu : Measure W) [SFinite mu]
(kappa : Kernel W X) [IsSFiniteKernel kappa]
(T : W ≃ᵐ W) {S : X → X} (hS : Measurable S)
(hT : MeasurePreserving T mu mu)
(hkappa : ∀ᵐ w ∂mu, kappa (T w) = (kappa w).map S) :
(mu ⊗ₘ kappa).map (Prod.map T S) = mu ⊗ₘ kappa := by
rw [map_compProd_prodMap mu kappa T hS, hT.map_eq]
apply Measure.compProd_congr
have htrans := hT.symm.quasiMeasurePreserving.ae hkappa
filter_upwards [htrans] with w hw
simpa only [Kernel.map_apply _ hS, Kernel.comap_apply,
MeasurableEquiv.apply_symm_apply] using hw.symm
lemma conditionalCopies_invariant
(mu : Measure W) [IsFiniteMeasure mu]
(kappa : Kernel W X) [IsFiniteKernel kappa]
[MeasurableSpace.CountableOrCountablyGenerated W X]
(T : W ≃ᵐ W) {S : X → X} (hS : Measurable S)
(hT : MeasurePreserving T mu mu)
(hjoint : (mu ⊗ₘ kappa).map (Prod.map T S) = mu ⊗ₘ kappa) :
(mu ⊗ₘ (kappa ×ₖ kappa)).map (Prod.map T (Prod.map S S)) =
mu ⊗ₘ (kappa ×ₖ kappa) := by
apply joint_invariant_of_kernel_ae_equivariant mu (kappa ×ₖ kappa) T
(hS.prodMap hS) hT
filter_upwards [kernel_ae_equivariant_of_joint_invariant mu kappa T hS hT hjoint]
with w hw
simp only [Kernel.prod_apply, hw]
exact Measure.map_prod_map (kappa w) (kappa w) hS hS
end Problem367
end
end OAI
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