| import OAI.Analysis.CircleDomains.Probability.CodeLawMeasures |
| import OAI.Analysis.CircleDomains.Probability.CodePathLaws |
| import OAI.Analysis.CircleDomains.Probability.ProbabilityMixtureMap |
|
|
| namespace OAI |
|
|
| /-! The mean occupation, tuple measure, and untagged path law of a code mixture |
| are the corresponding mixtures of the original means and path laws. -/ |
|
|
| open MeasureTheory |
| open scoped Topology |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| variable {X Y Z : Type*} [MetricSpace X] [MetricSpace Y] |
| [MeasurableSpace X] [MeasurableSpace C(PathTime, X)] [MeasurableSpace Z] |
|
|
| theorem measurable_codePathLaw (d : PathCodeData X Y) : |
| Measurable (codePathLaw d) := |
| measurable_probability_pushforward (codePath d) (measurable_codePath d) |
|
|
| theorem codePathLaw_probabilityMixture (d : PathCodeData X Y) |
| (P : ProbabilityMeasure Z) (law : Z → ProbabilityMeasure (BasePathCode d)) |
| (hlaw : Measurable law) : |
| codePathLaw d (probabilityMixture P law hlaw) = |
| probabilityMixture P (fun z => codePathLaw d (law z)) |
| ((measurable_codePathLaw d).comp hlaw) := |
| probabilityMixture_map P law hlaw (codePath d) (measurable_codePath d) |
|
|
| theorem measurable_meanCodeTuple (d : PathCodeData X Y) : |
| Measurable (meanCodeTuple d) := |
| (Measure.measurable_bind' (measurable_baseCodeTuple d)).comp measurable_subtype_coe |
|
|
| theorem meanCodeTuple_probabilityMixture (d : PathCodeData X Y) |
| (P : ProbabilityMeasure Z) (law : Z → ProbabilityMeasure (BasePathCode d)) |
| (hlaw : Measurable law) : |
| meanCodeTuple d (probabilityMixture P law hlaw) = |
| (P : Measure Z).bind (fun z => meanCodeTuple d (law z)) := by |
| exact Measure.bind_bind (measurable_subtype_coe.comp hlaw).aemeasurable |
| (measurable_baseCodeTuple d).aemeasurable |
|
|
| variable [MeasurableSpace Y] [BorelSpace Y] |
| [BorelSpace C(PathTime, X)] [MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] |
|
|
| theorem measurable_meanCodeOccupation (d : PathCodeData X Y) |
| (hU : IsOpen d.quotientDomain) : Measurable (meanCodeOccupation d) := |
| (Measure.measurable_bind' (measurable_baseCodeOccupation d hU)).comp measurable_subtype_coe |
|
|
| theorem meanCodeOccupation_probabilityMixture (d : PathCodeData X Y) |
| (hU : IsOpen d.quotientDomain) |
| (P : ProbabilityMeasure Z) (law : Z → ProbabilityMeasure (BasePathCode d)) |
| (hlaw : Measurable law) : |
| meanCodeOccupation d (probabilityMixture P law hlaw) = |
| (P : Measure Z).bind (fun z => meanCodeOccupation d (law z)) := by |
| exact Measure.bind_bind (measurable_subtype_coe.comp hlaw).aemeasurable |
| (measurable_baseCodeOccupation d hU).aemeasurable |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
