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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 | import Mathlib.Probability.Kernel.Condexp
import Mathlib.Tactic
namespace OAI
open MeasureTheory ProbabilityTheory
namespace Invariant233.CrossingAvoidance
variable {Ω β : Type*}
theorem ae_ae_condExpKernel_of_ae (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω}
[StandardBorelSpace Ω] (hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ]
{P : Ω → Prop} (hP : ∀ᵐ ω ∂μ, P ω) :
∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x, P ω := by
apply ae_of_ae_trim hm'
apply Measure.ae_ae_of_ae_comp
rwa [condExpKernel_comp_trim hm']
theorem condExpKernel_ae_eq_of_measurable (m' : MeasurableSpace Ω)
{mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω]
[mβ : MeasurableSpace β] [MeasurableEq β]
(hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ]
(f : Ω → β) (hf : Measurable[m'] f) :
∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x, f ω = f x := by
let : MeasurableSpace (Ω × Ω) := m'.prod mΩ
have hf1 : @Measurable (Ω × Ω) β (m'.prod mΩ) mβ (fun z => f z.1) :=
hf.comp (@measurable_fst Ω Ω m' mΩ)
have hf2 : @Measurable (Ω × Ω) β (m'.prod mΩ) mβ (fun z => f z.2) :=
(hf.mono hm' le_rfl).comp (@measurable_snd Ω Ω m' mΩ)
have hdiag : @Measurable Ω (Ω × Ω) mΩ (m'.prod mΩ) (fun x => (x, x)) :=
(measurable_id'' hm').prodMk measurable_id
have hprod : ∀ᵐ z ∂(μ.trim hm') ⊗ₘ condExpKernel μ m', f z.2 = f z.1 := by
rw [compProd_trim_condExpKernel hm']
exact (ae_map_iff hdiag.aemeasurable (measurableSet_eq_fun hf2 hf1)).mpr
(Filter.Eventually.of_forall (fun _ => rfl))
exact ae_of_ae_trim hm' (Measure.ae_ae_of_ae_compProd hprod)
end Invariant233.CrossingAvoidance
end OAI
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