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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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
variable {W Ω : Type*} [MeasurableSpace W] [MeasurableSpace Ω]
[StandardBorelSpace Ω] [Nonempty Ω]
{ν : Measure W} [IsFiniteMeasure ν] {κ : Kernel W Ω} [IsMarkovKernel κ]
theorem condExp_compProd_fst {f : W × Ω → ℝ} (hf : Integrable f (ν ⊗ₘ κ)) :
(ν ⊗ₘ κ)[f | (inferInstance : MeasurableSpace W).comap Prod.fst] =ᵐ[ν ⊗ₘ κ]
fun z => ∫ x, f (z.1, x) ∂κ z.1 := by
have hmap : (ν ⊗ₘ κ).map (fun z : W × Ω => (z.1, z.2)) =
(ν ⊗ₘ κ).map Prod.fst ⊗ₘ κ := by
change (ν ⊗ₘ κ).map id = (ν ⊗ₘ κ).fst ⊗ₘ κ
rw [Measure.map_id, Measure.fst_compProd]
have hk := condDistrib_ae_eq_of_measure_eq_compProd_of_measurable
(μ := ν ⊗ₘ κ) (κ := κ) measurable_fst measurable_snd hmap
have hc := condExp_prod_ae_eq_integral_condDistrib'
(μ := ν ⊗ₘ κ) (X := Prod.fst) (Y := Prod.snd) (f := f)
measurable_fst measurable_snd.aemeasurable
(by
change Integrable f ((ν ⊗ₘ κ).map id)
simpa only [Measure.map_id] using hf)
filter_upwards [hc, ae_of_ae_map measurable_fst.aemeasurable hk] with z hcz hkz
simpa only [Prod.mk.eta, hkz] using hcz
end Problem367
end
end OAI
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