| 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 |
|
|
