openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ConditionalEndpointTransport.lean
| import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | |
| import Mathlib.MeasureTheory.Function.FactorsThrough | |
| namespace OAI | |
| open MeasureTheory | |
| namespace Invariant233.ConditionalEndpointTransport | |
| variable {Ω X Y E : Type*} [mΩ : MeasurableSpace Ω] | |
| [mX : MeasurableSpace X] [mY : MeasurableSpace Y] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] | |
| theorem condExp_comap_encode_ae_eq | |
| (μ : Measure Ω) [IsFiniteMeasure μ] (x : Ω → X) (hx : Measurable x) | |
| (encode : X → Y) (hencode : Measurable encode) | |
| (decode : Y → X) (hdecode : Measurable decode) | |
| (hinverse : ∀ᵐ ω ∂μ, decode (encode (x ω)) = x ω) (f : Ω → E) : | |
| μ[f | mY.comap (encode ∘ x)] =ᵐ[μ] μ[f | mX.comap x] := by | |
| have hxy : mY.comap (encode ∘ x) ≤ mX.comap x := | |
| (hencode.comp (Measurable.of_comap_le le_rfl)).comap_le | |
| have hy : Measurable (encode ∘ x) := hencode.comp hx | |
| obtain ⟨g, hg, hfactor⟩ := | |
| (stronglyMeasurable_condExp (μ := μ) (f := f) (m := mX.comap x)).exists_eq_measurable_comp | |
| have hgm : AEStronglyMeasurable[mY.comap (encode ∘ x)] | |
| (μ[f | mX.comap x]) μ := by | |
| have hmeas : StronglyMeasurable[mY.comap (encode ∘ x)] | |
| (fun ω => g (decode (encode (x ω)))) := | |
| hg.comp_measurable (hdecode.comp (Measurable.of_comap_le le_rfl)) | |
| apply hmeas.aestronglyMeasurable.congr | |
| filter_upwards [hinverse] with ω hω | |
| rw [hω, hfactor] | |
| rfl | |
| exact (condExp_condExp_of_le hxy hx.comap_le).symm.trans | |
| (condExp_of_aestronglyMeasurable' hy.comap_le hgm integrable_condExp) | |
| end Invariant233.ConditionalEndpointTransport | |
| end OAI | |