File size: 1,838 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 | import Mathlib
namespace OAI
open MeasureTheory
namespace Problem367
variable {W X G : Type*} [MeasurableSpace W] [MeasurableSpace X]
theorem ae_equivariant_of_joint_recovery [MeasurableEq X]
{η : Measure (W × X)} {μ : Measure W}
{T : W → W} {S : X → X} {F : W → X}
(hμ : Measure.map Prod.fst η = μ)
(hT : Measurable T) (hS : Measurable S) (hF : Measurable F)
(hjoint : Measure.QuasiMeasurePreserving
(fun p : W × X => (T p.1, S p.2)) η η)
(hrecover : ∀ᵐ p ∂η, p.2 = F p.1) :
∀ᵐ w ∂μ, F (T w) = S (F w) := by
rw [← hμ]
apply (ae_map_iff measurable_fst.aemeasurable
(measurableSet_eq_fun (hF.comp hT) (hS.comp hF))).2
have htrans := hjoint.ae hrecover
filter_upwards [hrecover, htrans] with p hp hTp
exact hTp.symm.trans (congrArg S hp)
theorem ae_equivariant_family_of_joint_recovery [MeasurableEq X]
{η : Measure (W × X)} {μ : Measure W}
{T : G → W → W} {S : G → X → X} {F : W → X}
(hμ : Measure.map Prod.fst η = μ)
(hT : ∀ g, Measurable (T g)) (hS : ∀ g, Measurable (S g))
(hF : Measurable F)
(hjoint : ∀ g, Measure.QuasiMeasurePreserving
(fun p : W × X => (T g p.1, S g p.2)) η η)
(hrecover : ∀ᵐ p ∂η, p.2 = F p.1) :
∀ g, ∀ᵐ w ∂μ, F (T g w) = S g (F w) :=
fun g => ae_equivariant_of_joint_recovery hμ (hT g) (hS g) hF
(hjoint g) hrecover
theorem map_eq_second_marginal_of_joint_recovery
{η : Measure (W × X)} {μ : Measure W} {F : W → X}
(hμ : Measure.map Prod.fst η = μ) (hF : Measurable F)
(hrecover : ∀ᵐ p ∂η, p.2 = F p.1) :
Measure.map F μ = Measure.map Prod.snd η := by
rw [← hμ, Measure.map_map hF measurable_fst]
apply Measure.map_congr
filter_upwards [hrecover] with p hp
exact hp.symm
end Problem367
end OAI
|