openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ConditionalEventBridge.lean
| import Mathlib.Probability.Independence.Conditional | |
| import Mathlib.Probability.Independence.Basic | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped BigOperators | |
| namespace Invariant233.ConditionalEventBridge | |
| variable {Ω A B ι : Type*} | |
| [MeasurableSpace Ω] [StandardBorelSpace Ω] | |
| [MeasurableSpace A] [MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] | |
| [Countable ι] | |
| theorem iCondIndepSet_preimage_of_condDistrib | |
| (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (X : Ω → A) (Y : Ω → B) (hX : Measurable X) (hY : Measurable Y) | |
| (s : ι → Set B) (hs : ∀ i, MeasurableSet (s i)) | |
| (hind : ∀ᵐ x ∂μ.map X, iIndepSet s (condDistrib Y X μ x)) : | |
| iCondIndepSet (MeasurableSpace.comap X inferInstance) hX.comap_le | |
| (fun i => Y ⁻¹' s i) μ := by | |
| apply (iCondIndepSet_iff _ _ _ (fun i => hY (hs i)) μ).2 | |
| intro S | |
| have hi : ∀ᵐ ω ∂μ, ∀ i, | |
| (condDistrib Y X μ (X ω)).real (s i) = | |
| (μ[(Y ⁻¹' s i).indicator (fun _ => (1 : ℝ)) | | |
| MeasurableSpace.comap X inferInstance]) ω := | |
| ae_all_iff.mpr (fun i => condDistrib_ae_eq_condExp hX hY (hs i)) | |
| have hinter := condDistrib_ae_eq_condExp (μ := μ) hX hY | |
| (MeasurableSet.biInter S.countable_toSet (fun i _ => hs i)) | |
| have hind' := ae_of_ae_map hX.aemeasurable hind | |
| filter_upwards [hi, hinter, hind'] with ω hω hI hInd | |
| simp only [Set.preimage_iInter, Finset.mem_coe] at hI | |
| rw [← hI] | |
| have heq := congrArg ENNReal.toReal (hInd.meas_biInter S) | |
| simp only [ENNReal.toReal_prod] at heq | |
| change (condDistrib Y X μ (X ω)).real (⋂ i ∈ S, s i) = _ at heq | |
| rw [heq] | |
| simp only [Finset.prod_apply] | |
| exact Finset.prod_congr rfl (fun i _ => hω i) | |
| end Invariant233.ConditionalEventBridge | |
| end | |
| end OAI | |