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9eeb9c7 | 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 | import Mathlib.Probability.Independence.Conditional
import Mathlib.Probability.Independence.Basic
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace Invariant233.ConditionalEventBridge
variable {Ω A B ι : Type*}
[MeasurableSpace Ω] [StandardBorelSpace Ω]
[MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace B] [Nonempty B]
[Countable ι]
theorem iCondIndepSet_preimage_of_condDistrib
(μ : Measure Ω) [IsFiniteMeasure μ]
(X : Ω → A) (Y : Ω → B) (hX : Measurable X) (hY : Measurable Y)
(s : ι → Set B) (hs : ∀ i, MeasurableSet (s i))
(hind : ∀ᵐ x ∂μ.map X, iIndepSet s (condDistrib Y X μ x)) :
iCondIndepSet (MeasurableSpace.comap X inferInstance) hX.comap_le
(fun i => Y ⁻¹' s i) μ := by
apply (iCondIndepSet_iff _ _ _ (fun i => hY (hs i)) μ).2
intro S
have hi : ∀ᵐ ω ∂μ, ∀ i,
(condDistrib Y X μ (X ω)).real (s i) =
(μ[(Y ⁻¹' s i).indicator (fun _ => (1 : ℝ)) |
MeasurableSpace.comap X inferInstance]) ω :=
ae_all_iff.mpr (fun i => condDistrib_ae_eq_condExp hX hY (hs i))
have hinter := condDistrib_ae_eq_condExp (μ := μ) hX hY
(MeasurableSet.biInter S.countable_toSet (fun i _ => hs i))
have hind' := ae_of_ae_map hX.aemeasurable hind
filter_upwards [hi, hinter, hind'] with ω hω hI hInd
simp only [Set.preimage_iInter, Finset.mem_coe] at hI
rw [← hI]
have heq := congrArg ENNReal.toReal (hInd.meas_biInter S)
simp only [ENNReal.toReal_prod] at heq
change (condDistrib Y X μ (X ω)).real (⋂ i ∈ S, s i) = _ at heq
rw [heq]
simp only [Finset.prod_apply]
exact Finset.prod_congr rfl (fun i _ => hω i)
end Invariant233.ConditionalEventBridge
end
end OAI
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