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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 | import OAI.Analysis.CircleDomains.Sobolev.LipschitzArclengthBounds
import OAI.Analysis.CircleDomains.Probability.PathOccupation
namespace OAI
/-! Finite weighted Hausdorff length pays for an actual embedded path
whenever the chosen parameter is Lipschitz in both directions. -/
open Set Function MeasureTheory Metric
open scoped ENNReal NNReal
noncomputable section
namespace CircleDomainRigidity
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
theorem bilipschitz_pathOccupation_le (γ : RectifiablePath X) {L K : ℝ≥0}
(hL : LipschitzWith L γ.val) (hL0 : L ≠ 0)
(hK : AntilipschitzWith K γ.val) :
pathOccupation γ ≤ ((L : ℝ≥0∞) * K) •
(μH[1] : Measure X).restrict (range γ.val) := by
apply Measure.le_iff.mpr
intro s hs
rw [pathOccupation_apply γ hs]
have he : γ.val ⁻¹' s = γ.val ⁻¹' (s ∩ range γ.val) := by
ext t
simp only [mem_preimage, mem_inter_iff]
exact ⟨fun h => ⟨h, mem_range_self t⟩, And.left⟩
calc
pathLengthMeasure γ (γ.val ⁻¹' s) ≤
(L : ℝ≥0∞) * (μH[1] : Measure PathTime) (γ.val ⁻¹' s) :=
lipschitz_pathLength_le_hausdorff γ hL hL0 _
_ = (L : ℝ≥0∞) * (μH[1] : Measure PathTime)
(γ.val ⁻¹' (s ∩ range γ.val)) := by rw [he]
_ ≤ (L : ℝ≥0∞) * ((K : ℝ≥0∞) * (μH[1] : Measure X) (s ∩ range γ.val)) := by
apply mul_le_mul' le_rfl
simpa only [ENNReal.rpow_one] using
hK.hausdorffMeasure_preimage_le (show (0 : ℝ) ≤ 1 by norm_num) (s ∩ range γ.val)
_ = _ := by rw [Measure.smul_apply, smul_eq_mul, Measure.restrict_apply hs, mul_assoc]
theorem bilipschitz_pathCost_lt_top (γ : RectifiablePath X) {L K : ℝ≥0}
(hL : LipschitzWith L γ.val) (hL0 : L ≠ 0)
(hK : AntilipschitzWith K γ.val) (g : X → ℝ≥0∞)
(hfinite : (∫⁻ x in range γ.val, g x ∂(μH[1] : Measure X)) < ∞) :
(∫⁻ x, g x ∂pathOccupation γ) < ∞ := by
calc
(∫⁻ x, g x ∂pathOccupation γ) ≤
∫⁻ x, g x ∂(((L : ℝ≥0∞) * K) •
(μH[1] : Measure X).restrict (range γ.val)) :=
lintegral_mono' (bilipschitz_pathOccupation_le γ hL hL0 hK) (fun _ => le_rfl)
_ = ((L : ℝ≥0∞) * K) *
∫⁻ x in range γ.val, g x ∂(μH[1] : Measure X) := lintegral_smul_measure _ _
_ < ∞ := ENNReal.mul_lt_top (ENNReal.mul_lt_top ENNReal.coe_lt_top ENNReal.coe_lt_top) hfinite
end CircleDomainRigidity
end
end OAI
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