| 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 | |