File size: 2,901 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 65 66 | 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
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