File size: 2,639 Bytes
9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | 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
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