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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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.Analysis.CircleDomains.Probability.PathCostMeasure
namespace OAI
/-! Actual mean occupation, atomic and tuple measures of a law on base
codes. Their sum is exactly the barycenter of the coding measures. -/
open Set MeasureTheory
open scoped ENNReal
noncomputable section
namespace CircleDomainRigidity
variable {X Y D : Type*} [MetricSpace X] [MetricSpace Y] [TopologicalSpace D]
[MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y]
[MeasurableSpace D] [BorelSpace D]
[MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)]
[MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)]
[SecondCountableTopology X] [Countable D]
omit [TopologicalSpace D] [MeasurableSpace Y] [BorelSpace Y] [BorelSpace D]
[MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] in
theorem measurable_baseCodeAtomic (d : PathCodeData X Y) (p : D → X) (c : D → ℝ≥0∞) :
Measurable (fun Γ : BasePathCode d => atomicVisitMeasure p c Γ.val.1) :=
(measurable_atomicVisitMeasure p c).comp (measurable_fst.comp measurable_subtype_coe)
omit [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y]
[BorelSpace C(PathTime, X)] [MeasurableSpace C(PathTime, Y)]
[BorelSpace C(PathTime, Y)] [SecondCountableTopology X] in
theorem measurable_baseCodeTuple (d : PathCodeData X Y) :
Measurable (fun Γ : BasePathCode d => weightedTupleMeasure Γ.val.2) :=
measurable_weightedTupleMeasure.comp (measurable_snd.comp measurable_subtype_coe)
def meanCodeOccupation (d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) : Measure Y :=
(P : Measure (BasePathCode d)).bind (baseCodeOccupation d)
def meanCodeAtomic (d : PathCodeData X Y) (p : D → X) (c : D → ℝ≥0∞)
(P : ProbabilityMeasure (BasePathCode d)) : Measure D :=
(P : Measure (BasePathCode d)).bind (fun Γ => atomicVisitMeasure p c Γ.val.1)
def meanCodeTuple (d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) :
Measure LevelTuple :=
(P : Measure (BasePathCode d)).bind (fun Γ => weightedTupleMeasure Γ.val.2)
def meanCodeCost (d : PathCodeData X Y) (p : D → X) (c : D → ℝ≥0∞)
(P : ProbabilityMeasure (BasePathCode d)) : Measure (PathCostSpace Y D) :=
(P : Measure (BasePathCode d)).bind (pathCostMeasure d p c)
omit [BorelSpace X] [SecondCountableTopology X] in
theorem meanCodeOccupation_apply (d : PathCodeData X Y) (hU : IsOpen d.quotientDomain)
(P : ProbabilityMeasure (BasePathCode d)) {A : Set Y} (hA : MeasurableSet A) :
meanCodeOccupation d P A = ∫⁻ Γ, baseCodeOccupation d Γ A ∂(P : Measure (BasePathCode d)) :=
Measure.bind_apply hA (measurable_baseCodeOccupation d hU).aemeasurable
omit [TopologicalSpace D] [MeasurableSpace Y] [BorelSpace Y] [BorelSpace D]
[MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] in
theorem meanCodeAtomic_apply (d : PathCodeData X Y) (p : D → X) (c : D → ℝ≥0∞)
(P : ProbabilityMeasure (BasePathCode d)) {A : Set D} (hA : MeasurableSet A) :
meanCodeAtomic d p c P A = ∫⁻ Γ, atomicVisitMeasure p c Γ.val.1 A ∂(P : Measure (BasePathCode d)) :=
Measure.bind_apply hA (measurable_baseCodeAtomic d p c).aemeasurable
omit [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y]
[BorelSpace C(PathTime, X)] [MeasurableSpace C(PathTime, Y)]
[BorelSpace C(PathTime, Y)] [SecondCountableTopology X] in
theorem meanCodeTuple_apply (d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d))
{A : Set LevelTuple} (hA : MeasurableSet A) :
meanCodeTuple d P A = ∫⁻ Γ, weightedTupleMeasure Γ.val.2 A ∂(P : Measure (BasePathCode d)) :=
Measure.bind_apply hA (measurable_baseCodeTuple d).aemeasurable
theorem meanCodeCost_apply (d : PathCodeData X Y) (hU : IsOpen d.quotientDomain)
(p : D → X) (c : D → ℝ≥0∞) (P : ProbabilityMeasure (BasePathCode d))
{A : Set (PathCostSpace Y D)} (hA : MeasurableSet A) :
meanCodeCost d p c P A = ∫⁻ Γ, pathCostMeasure d p c Γ A ∂(P : Measure (BasePathCode d)) :=
Measure.bind_apply hA (measurable_pathCostMeasure d hU p c).aemeasurable
theorem meanCodeCost_eq (d : PathCodeData X Y) (hU : IsOpen d.quotientDomain)
(p : D → X) (c : D → ℝ≥0∞) (P : ProbabilityMeasure (BasePathCode d)) :
meanCodeCost d p c P =
(meanCodeOccupation d P).map costPhysical +
(meanCodeAtomic d p c P).map costAtomic + (meanCodeTuple d P).map costTuple := by
ext A hA
rw [meanCodeCost_apply d hU p c P hA, Measure.add_apply, Measure.add_apply,
Measure.map_apply measurable_costPhysical hA, Measure.map_apply measurable_costAtomic hA,
Measure.map_apply measurable_costTuple hA,
meanCodeOccupation_apply d hU P (measurable_costPhysical hA),
meanCodeAtomic_apply d p c P (measurable_costAtomic hA),
meanCodeTuple_apply d P (measurable_costTuple hA)]
simp_rw [pathCostMeasure, Measure.add_apply,
Measure.map_apply measurable_costPhysical hA, Measure.map_apply measurable_costAtomic hA,
Measure.map_apply measurable_costTuple hA]
have hp : Measurable (fun Γ : BasePathCode d =>
baseCodeOccupation d Γ (costPhysical ⁻¹' A)) :=
(Measure.measurable_coe (measurable_costPhysical hA)).comp
(measurable_baseCodeOccupation d hU)
have ha : Measurable (fun Γ : BasePathCode d =>
atomicVisitMeasure p c Γ.val.1 (costAtomic ⁻¹' A)) :=
(Measure.measurable_coe (measurable_costAtomic hA)).comp
(measurable_baseCodeAtomic d p c)
have hpa : Measurable (fun Γ : BasePathCode d =>
baseCodeOccupation d Γ (costPhysical ⁻¹' A) +
atomicVisitMeasure p c Γ.val.1 (costAtomic ⁻¹' A)) := hp.add ha
rw [lintegral_add_left hpa, lintegral_add_left hp]
end CircleDomainRigidity
end
end OAI
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