File size: 4,259 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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
theorem condExp_mul_of_condIndepFun {Ω : Type*} [mΩ : MeasurableSpace Ω]
[StandardBorelSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
{m : MeasurableSpace Ω} (hm : m ≤ mΩ) {f g : Ω → ℝ}
(hind : CondIndepFun m hm f g μ) (hf : Measurable[mΩ] f) (hg : Measurable[mΩ] g)
(hfi : Integrable f μ) (hgi : Integrable g μ)
(hfgi : Integrable (fun ω => f ω * g ω) μ) :
μ[(fun ω => f ω * g ω) | m] =ᵐ[μ] fun ω => μ[f | m] ω * μ[g | m] ω := by
let : MeasurableSpace Ω := mΩ
have hmap := ae_of_ae_trim (μ := μ) hm
((condIndepFun_iff_map_prod_eq_prod_map_map (mΩ := mΩ) (m' := m)
(hm' := hm) (μ := μ) hf hg).mp hind)
filter_upwards [condExp_ae_eq_integral_condExpKernel (mΩ := mΩ) (μ := μ) hm hfgi,
condExp_ae_eq_integral_condExpKernel (mΩ := mΩ) (μ := μ) hm hfi,
condExp_ae_eq_integral_condExpKernel (mΩ := mΩ) (μ := μ) hm hgi, hmap] with ω hfg hfω hgω hmapω
rw [hfg, hfω, hgω]
have hmap' : Measure.map (fun y => (f y, g y)) (condExpKernel μ m ω) =
(Measure.map f (condExpKernel μ m ω)).prod (Measure.map g (condExpKernel μ m ω)) := by
simpa only [Kernel.prod_apply, Kernel.map_apply _ (hf.prodMk hg),
Kernel.map_apply _ hf, Kernel.map_apply _ hg] using hmapω
have hmfun : Measurable (fun p : ℝ × ℝ => p.1 * p.2) := measurable_fst.mul measurable_snd
rw [← integral_map (hf.prodMk hg).aemeasurable hmfun.aestronglyMeasurable, hmap',
integral_prod_mul (fun x : ℝ => x) (fun y : ℝ => y)]
congr 1
· simpa only [id_eq] using integral_map (μ := condExpKernel μ m ω)
hf.aemeasurable measurable_id.aestronglyMeasurable
· simpa only [id_eq] using integral_map (μ := condExpKernel μ m ω)
hg.aemeasurable measurable_id.aestronglyMeasurable
theorem integral_mul_eq_integral_condExp_mul_of_condIndepFun {Ω : Type*}
[mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
{f g : Ω → ℝ} (hind : CondIndepFun m hm f g μ)
(hf : Measurable[mΩ] f) (hg : Measurable[mΩ] g) (hfi : Integrable f μ) (hgi : Integrable g μ)
(hfgi : Integrable (fun ω => f ω * g ω) μ) :
(∫ ω, f ω * g ω ∂μ) = ∫ ω, μ[f | m] ω * μ[g | m] ω ∂μ := by
let : MeasurableSpace Ω := mΩ
rw [← integral_condExp (m₀ := mΩ) (μ := μ) hm (f := fun ω => f ω * g ω)]
exact integral_congr_ae (μ := μ)
(condExp_mul_of_condIndepFun (mΩ := mΩ) (μ := μ) hm hind hf hg hfi hgi hfgi)
theorem condExp_pair_eq_of_condIndepFun {Ω Z W : Type*}
[MeasurableSpace Ω] [StandardBorelSpace Ω] [mZ : MeasurableSpace Z]
[mW : MeasurableSpace W] [StandardBorelSpace W] [Nonempty W]
{μ : Measure Ω} [IsProbabilityMeasure μ] {z : Ω → Z} {w : Ω → W} {f : Ω → ℝ}
(hz : Measurable z) (hw : Measurable w) (hf : Measurable f) (hfi : Integrable f μ)
(hind : CondIndepFun (mZ.comap z) hz.comap_le w f μ) :
μ[f | (inferInstance : MeasurableSpace (Z × W)).comap (fun ω => (z ω, w ω))] =ᵐ[μ] μ[f | mZ.comap z] := by
have hdist := (condIndepFun_iff_condDistrib_prod_ae_eq_prodMkRight hf hw hz).mp hind
have hpull := ae_of_ae_map (hz.prodMk hw).aemeasurable hdist
filter_upwards [condExp_ae_eq_integral_condDistrib' (hz.prodMk hw) hfi,
condExp_ae_eq_integral_condDistrib' hz hfi, hpull] with ω hpair hzω hdistω
rw [hpair, hzω, hdistω, Kernel.prodMkRight_apply]
theorem condExp_sup_eq_of_condIndepFun {Ω Z W : Type*}
[MeasurableSpace Ω] [StandardBorelSpace Ω] [mZ : MeasurableSpace Z]
[mW : MeasurableSpace W] [StandardBorelSpace W] [Nonempty W]
{μ : Measure Ω} [IsProbabilityMeasure μ] {z : Ω → Z} {w : Ω → W} {f : Ω → ℝ}
(hz : Measurable z) (hw : Measurable w) (hf : Measurable f) (hfi : Integrable f μ)
(hind : CondIndepFun (mZ.comap z) hz.comap_le w f μ) :
μ[f | mZ.comap z ⊔ mW.comap w] =ᵐ[μ] μ[f | mZ.comap z] := by
simpa only [Prod.instMeasurableSpace, MeasurableSpace.comap_prodMk] using
condExp_pair_eq_of_condIndepFun hz hw hf hfi hind
end Problem367
end
end OAI
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