openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ConditionalDistributionEvents.lean
| import Mathlib.Probability.Independence.Conditional | |
| import Mathlib.Probability.Independence.Basic | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped BigOperators ProbabilityTheory | |
| namespace Invariant233.ConditionalDistributionEvents | |
| variable {ι Ω X Y : Type*} [Countable ι] | |
| [MeasurableSpace Ω] [StandardBorelSpace Ω] | |
| [MeasurableSpace X] [MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y] | |
| theorem iCondIndepSet_preimage_of_condDistrib | |
| (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (observed : Ω → X) (value : Ω → Y) | |
| (hobserved : Measurable observed) (hvalue : Measurable value) | |
| (events : ι → Set Y) (hevents : ∀ i, MeasurableSet (events i)) | |
| (hind : ∀ᵐ x ∂μ.map observed, | |
| iIndepSet events (condDistrib value observed μ x)) : | |
| iCondIndepSet (MeasurableSpace.comap observed inferInstance) hobserved.comap_le | |
| (fun i => value ⁻¹' events i) μ := by | |
| apply (iCondIndepSet_iff _ _ _ (fun i => hvalue (hevents i)) μ).mpr | |
| intro S | |
| have hi : MeasurableSet (⋂ i ∈ S, events i) := | |
| MeasurableSet.biInter S.countable_toSet (fun i _ => hevents i) | |
| have hfull := ae_of_ae_map hobserved.aemeasurable hind | |
| have heach : ∀ᵐ ω ∂μ, ∀ i, | |
| (condDistrib value observed μ (observed ω)).real (events i) = | |
| (μ⟦value ⁻¹' events i | MeasurableSpace.comap observed inferInstance⟧) ω := | |
| ae_all_iff.mpr (fun i => condDistrib_ae_eq_condExp hobserved hvalue (hevents i)) | |
| filter_upwards [hfull, heach, condDistrib_ae_eq_condExp hobserved hvalue hi] | |
| with ω hω heach hinter | |
| have hprod := (iIndepSet_iff_meas_biInter hevents).mp hω S | |
| have hreal := congrArg ENNReal.toReal hprod | |
| simp only [ENNReal.toReal_prod] at hreal | |
| have hpre : value ⁻¹' (⋂ i ∈ S, events i) = ⋂ i ∈ S, value ⁻¹' events i := by | |
| ext z | |
| simp | |
| rw [hpre] at hinter | |
| simp only [Finset.prod_apply] | |
| rw [← hinter] | |
| change (condDistrib value observed μ (observed ω) (⋂ i ∈ S, events i)).toReal = _ | |
| rw [hreal] | |
| exact Finset.prod_congr rfl (fun i _ => heach i) | |
| theorem iCondIndepSet_coordinate_preimage_of_condDistrib | |
| {Z : ι → Type*} [∀ i, MeasurableSpace (Z i)] | |
| (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (observed : Ω → X) (value : Ω → Y) | |
| (hobserved : Measurable observed) (hvalue : Measurable value) | |
| (coordinate : ∀ i, Y → Z i) (hcoordinate : ∀ i, Measurable (coordinate i)) | |
| (events : ∀ i, Set (Z i)) (hevents : ∀ i, MeasurableSet (events i)) | |
| (hind : ∀ᵐ x ∂μ.map observed, | |
| iIndepFun coordinate (condDistrib value observed μ x)) : | |
| iCondIndepSet (MeasurableSpace.comap observed inferInstance) hobserved.comap_le | |
| (fun i => (fun ω => coordinate i (value ω)) ⁻¹' events i) μ := by | |
| apply iCondIndepSet_preimage_of_condDistrib μ observed value hobserved hvalue | |
| (fun i => coordinate i ⁻¹' events i) (fun i => hcoordinate i (hevents i)) | |
| filter_upwards [hind] with x hx | |
| apply (iIndepSet_iff_meas_biInter (fun i => hcoordinate i (hevents i))).mpr | |
| intro S | |
| exact hx.measure_inter_preimage_eq_mul S (fun i _ => hevents i) | |
| end Invariant233.ConditionalDistributionEvents | |
| end | |
| end OAI | |