| import OAI.Analysis.CircleDomains.Modulus.CountablePhysicalLength |
| import OAI.Analysis.CircleDomains.Probability.CollarTimeMeasures |
| import OAI.Analysis.CircleDomains.Probability.PhysicalPieceTimeMeasure |
|
|
| namespace OAI |
|
|
| /-! Exact physical time and weighted occupation measures for collar |
| concatenations. Parameter multiplicity is preserved. -/ |
|
|
| open Set Function MeasureTheory |
| open scoped ENNReal |
|
|
| noncomputable section |
|
|
| namespace CircleDomainRigidity |
|
|
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] {G : Set Y} {U : Set X} |
|
|
| theorem physicalTimeLength_collarPieces (e : G ≃ₜ U) (hU : IsOpen U) (y₀ : Y) |
| (η : FinitePhysicalPath e y₀) (γ : ℕ → FinitePhysicalPath e y₀) |
| (hmatch : ∀ n s, η.val (affinePathTime (collarTime (n + 1)) (collarTime n) s) = (γ n).val s) : |
| physicalTimeLength e y₀ η = Measure.sum (fun n => |
| (physicalTimeLength e y₀ (γ n)).map (affinePathTime (collarTime (n + 1)) (collarTime n))) := by |
| rw [← sum_collar_restrict (physicalTimeLength e y₀ η)] |
| congr 1 |
| funext n |
| exact physicalTimeLength_piece e hU y₀ η (γ n) |
| (collarTime_strictAnti (Nat.lt_succ_self n)) (hmatch n) |
|
|
| theorem physicalDomainLength_collarPieces (e : G ≃ₜ U) (hU : IsOpen U) (y₀ : Y) |
| (η : FinitePhysicalPath e y₀) (γ : ℕ → FinitePhysicalPath e y₀) |
| (hmatch : ∀ n s, η.val (affinePathTime (collarTime (n + 1)) (collarTime n) s) = (γ n).val s) : |
| physicalDomainLength e y₀ η.val = ∑' n, physicalDomainLength e y₀ (γ n).val := by |
| rw [← physicalTimeLength_univ e y₀ η, physicalTimeLength_collarPieces e hU y₀ η γ hmatch, |
| Measure.sum_apply _ MeasurableSet.univ] |
| apply tsum_congr |
| intro n |
| rw [Measure.map_apply (affinePathTime _ _).continuous.measurable MeasurableSet.univ, |
| preimage_univ, physicalTimeLength_univ] |
|
|
| variable [MeasurableSpace X] [BorelSpace X] [MeasurableSpace Y] [BorelSpace Y] |
|
|
| theorem physicalCost_collarPieces (e : G ≃ₜ U) (hU : IsOpen U) (y₀ : Y) |
| (η : FinitePhysicalPath e y₀) (γ : ℕ → FinitePhysicalPath e y₀) |
| (hmatch : ∀ n s, η.val (affinePathTime (collarTime (n + 1)) (collarTime n) s) = (γ n).val s) |
| (g : Y → ℝ≥0∞) (hg : Measurable g) : |
| (∫⁻ y, g y ∂dyadicPhysicalOccupation e y₀ η) = |
| ∑' n, ∫⁻ y, g y ∂dyadicPhysicalOccupation e y₀ (γ n) := by |
| rw [physicalOccupation_integral e hU y₀ η g hg, |
| physicalTimeLength_collarPieces e hU y₀ η γ hmatch, lintegral_sum_measure] |
| apply tsum_congr |
| intro n |
| have hmeas : Measurable (fun t : PathTime => g (physicalValue e y₀ (η.val t))) := |
| hg.comp ((measurable_physicalValue e hU y₀).comp η.val.continuous.measurable) |
| rw [lintegral_map hmeas |
| (affinePathTime _ _).continuous.measurable, physicalOccupation_integral e hU y₀ (γ n) g hg] |
| apply lintegral_congr |
| intro t |
| simp only [hmatch] |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
