openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ConditionalObservationTransport.lean
| 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 | |