| import OAI.Analysis.FiniteFactor.FiniteProductAbsoluteContinuity |
| import Mathlib.Algebra.BigOperators.Group.Finset.Powerset |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Finset |
| open scoped ENNReal |
| noncomputable section |
|
|
| / |
| def probabilityAverage {X : Type*} [MeasurableSpace X] (μ ν : Measure X) : Measure X := |
| (2 : ℝ≥0∞)⁻¹ • (μ + ν) |
| |
| instance probabilityAverage_probability {X : Type*} [MeasurableSpace X] |
| (μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] : |
| IsProbabilityMeasure (probabilityAverage μ ν) := by |
| constructor |
| simpa [probabilityAverage,Measure.smul_apply,Measure.add_apply,measure_univ] |
| using ENNReal.inv_two_add_inv_two |
| |
| / |
| def coordinateUpdateLaw {ι X : Type*} [DecidableEq ι] [MeasurableSpace X] |
| (i : ι) (μ : Measure X) (ν : Measure (ι → X)) : Measure (ι → X) := |
| (μ.prod ν).map (fun z : X × (ι → X) => Function.update z.2 i z.1) |
| |
| theorem measurable_coordinateUpdate {ι X : Type*} [DecidableEq ι] [MeasurableSpace X] (i : ι) : |
| Measurable (fun z : X × (ι → X) => Function.update z.2 i z.1) := by |
| apply Measurable.of_eval |
| intro k |
| by_cases hk : k = i |
| · subst k; simpa using (measurable_fst : Measurable (Prod.fst : X × (ι → X) → X)) |
| · simpa [Function.update_of_ne hk,Function.comp_def] using (measurable_pi_apply k).comp |
| (measurable_snd : Measurable (Prod.snd : X × (ι → X) → (ι → X))) |
| |
| theorem coordinateUpdateLaw_add_right {ι X : Type*} [DecidableEq ι] [MeasurableSpace X] |
| (i : ι) (μ : Measure X) (ν η : Measure (ι → X)) |
| [SFinite μ] [SFinite ν] [SFinite η] : |
| coordinateUpdateLaw i μ (ν+η) = coordinateUpdateLaw i μ ν + coordinateUpdateLaw i μ η := by |
| rw [coordinateUpdateLaw,Measure.prod_add,Measure.map_add _ _ (measurable_coordinateUpdate i)] |
| rfl |
| |
| theorem coordinateUpdateLaw_average {ι X : Type*} [DecidableEq ι] [MeasurableSpace X] |
| (i : ι) (μ ν : Measure X) (η : Measure (ι → X)) |
| [SFinite μ] [SFinite ν] [SFinite η] : |
| coordinateUpdateLaw i (probabilityAverage μ ν) η = |
| (2 : ℝ≥0∞)⁻¹ • (coordinateUpdateLaw i μ η + coordinateUpdateLaw i ν η) := by |
| unfold coordinateUpdateLaw probabilityAverage |
| rw [Measure.prod_smul_left,Measure.add_prod, |
| Measure.map_smul _ (measurable_coordinateUpdate i).aemeasurable, |
| Measure.map_add _ _ (measurable_coordinateUpdate i)] |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
