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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 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
import Mathlib.MeasureTheory.Function.FactorsThrough
import Mathlib.Probability.Independence.Conditional
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped MeasureTheory
namespace Invariant233.ConditionalObservationTransport
variable {Ω D Y E : Type*} [mΩ : MeasurableSpace Ω]
[mD : MeasurableSpace D] [mY : MeasurableSpace Y]
{μ : Measure Ω} [IsFiniteMeasure μ]
omit [IsFiniteMeasure μ] in
theorem aestronglyMeasurable_encoding
[NormedAddCommGroup E] [CompleteSpace E]
{X : Ω → D} {encode : D → Y} {decode : Y → D}
(hdecode : Measurable decode)
(hrecover : ∀ᵐ ω ∂μ, decode (encode (X ω)) = X ω)
{g : Ω → E} (hg : StronglyMeasurable[mD.comap X] g) :
AEStronglyMeasurable[mY.comap (encode ∘ X)] g μ := by
obtain ⟨k, hk, hkg⟩ := hg.exists_eq_measurable_comp
have hobs : Measurable[mY.comap (encode ∘ X)] (encode ∘ X) :=
measurable_iff_comap_le.mpr le_rfl
refine ⟨k ∘ decode ∘ (encode ∘ X),
hk.comp_measurable (hdecode.comp hobs), ?_⟩
filter_upwards [hrecover] with ω hω
simp only [hkg, Function.comp_apply, hω]
theorem condExp_encoding
[NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
{X : Ω → D} {encode : D → Y} {decode : Y → D}
(hX : Measurable X) (hencode : Measurable encode) (hdecode : Measurable decode)
(hrecover : ∀ᵐ ω ∂μ, decode (encode (X ω)) = X ω) (f : Ω → E) :
μ[f | mY.comap (encode ∘ X)] =ᵐ[μ] μ[f | mD.comap X] := by
have hle : mY.comap (encode ∘ X) ≤ mD.comap X := by
rw [← MeasurableSpace.comap_comp]
exact MeasurableSpace.comap_mono hencode.comap_le
have hsmall : mY.comap (encode ∘ X) ≤ mΩ := (hencode.comp hX).comap_le
have hg : AEStronglyMeasurable[mY.comap (encode ∘ X)] (μ[f | mD.comap X]) μ :=
aestronglyMeasurable_encoding hdecode hrecover stronglyMeasurable_condExp
exact (condExp_condExp_of_le hle hX.comap_le).symm.trans
(condExp_of_aestronglyMeasurable' hsmall hg integrable_condExp)
theorem condExp_encoding_formula
[NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
{X : Ω → D} {encode : D → Y} {decode : Y → D}
(hX : Measurable X) (hencode : Measurable encode) (hdecode : Measurable decode)
(hrecover : ∀ᵐ ω ∂μ, decode (encode (X ω)) = X ω)
(f : Ω → E) (posterior : D → E)
(hposterior : μ[f | mD.comap X] =ᵐ[μ] posterior ∘ X) :
μ[f | mY.comap (encode ∘ X)] =ᵐ[μ] posterior ∘ decode ∘ (encode ∘ X) := by
refine (condExp_encoding hX hencode hdecode hrecover f).trans (hposterior.trans ?_)
filter_upwards [hrecover] with ω hω
simp only [Function.comp_apply, hω]
theorem iCondIndepFun_encoding_iff
[StandardBorelSpace Ω] {ι : Type*} {β : ι → Type*}
[mβ : ∀ i, MeasurableSpace (β i)]
{X : Ω → D} {encode : D → Y} {decode : Y → D}
(hX : Measurable X) (hencode : Measurable encode) (hdecode : Measurable decode)
(hrecover : ∀ᵐ ω ∂μ, decode (encode (X ω)) = X ω)
(L : ∀ i, Ω → β i) (hL : ∀ i, Measurable (L i)) :
iCondIndepFun (mY.comap (encode ∘ X)) (hencode.comp hX).comap_le L μ ↔
iCondIndepFun (mD.comap X) hX.comap_le L μ := by
classical
rw [iCondIndepFun_iff_condExp_inter_preimage_eq_mul mβ L hL,
iCondIndepFun_iff_condExp_inter_preimage_eq_mul mβ L hL]
have hce (s : Set Ω) :
μ⟦s | mY.comap (encode ∘ X)⟧ =ᵐ[μ] μ⟦s | mD.comap X⟧ :=
condExp_encoding hX hencode hdecode hrecover (s.indicator (fun _ => (1 : ℝ)))
constructor
· intro h S sets hsets
have hp : (∏ i ∈ S, μ⟦L i ⁻¹' sets i | mY.comap (encode ∘ X)⟧) =ᵐ[μ]
∏ i ∈ S, μ⟦L i ⁻¹' sets i | mD.comap X⟧ :=
eventuallyEq_prod (fun i _ => hce (L i ⁻¹' sets i))
exact (hce _).symm.trans ((h S hsets).trans hp)
· intro h S sets hsets
have hp : (∏ i ∈ S, μ⟦L i ⁻¹' sets i | mY.comap (encode ∘ X)⟧) =ᵐ[μ]
∏ i ∈ S, μ⟦L i ⁻¹' sets i | mD.comap X⟧ :=
eventuallyEq_prod (fun i _ => hce (L i ⁻¹' sets i))
exact (hce _).trans ((h S hsets).trans hp.symm)
end Invariant233.ConditionalObservationTransport
end
end OAI
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