| import OAI.Analysis.CircleDomains.Probability.OccupationProduct |
| import OAI.Analysis.CircleDomains.Topology.BallPairKernel |
|
|
| namespace OAI |
|
|
| /-! The proximity integral of independent physical path occupations is |
| controlled by their local square energies. -/ |
|
|
| open Set Function Filter MeasureTheory |
| open scoped ENNReal NNReal Topology |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] |
| [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace Y] [SecondCountableTopology Y] |
| [MeasurableSpace C(PathTime, X)] [BorelSpace C(PathTime, X)] |
| [MeasurableSpace C(PathTime, Y)] [BorelSpace C(PathTime, Y)] |
|
|
| def codePairProximity (d e : PathCodeData X Y) (A : Set Y) (a : ℝ≥0) |
| (p : BasePathCode d × BasePathCode e) : ℝ≥0∞ := |
| ∫⁻ z, ballPairKernel A a z ∂(baseCodeOccupation d p.1).prod (baseCodeOccupation e p.2) |
|
|
| theorem measurable_codePairProximity (d e : PathCodeData X Y) |
| (hd : IsOpen d.quotientDomain) (he : IsOpen e.quotientDomain) |
| (A : Set Y) (hA : MeasurableSet A) (a : ℝ≥0) : |
| Measurable (codePairProximity d e A a) := |
| (Measure.measurable_lintegral (measurable_ballPairKernel A hA a)).comp |
| (measurable_finite_measure_product _ _ (measurable_baseCodeOccupation d hd) |
| (measurable_baseCodeOccupation e he)) |
|
|
| theorem mean_codePairProximity_le (d e : PathCodeData X Y) |
| (hd : IsOpen d.quotientDomain) (he : IsOpen e.quotientDomain) |
| (P : ProbabilityMeasure (BasePathCode d)) (Q : ProbabilityMeasure (BasePathCode e)) |
| [SigmaFinite (meanCodeOccupation d P)] [SigmaFinite (meanCodeOccupation e Q)] |
| (σ : Measure Y) [SFinite σ] (g h : Y → ℝ≥0∞) |
| (hg : Measurable g) (hh : Measurable h) |
| (hP : meanCodeOccupation d P = σ.withDensity g) |
| (hQ : meanCodeOccupation e Q = σ.withDensity h) |
| (A : Set Y) (hA : MeasurableSet A) {a : ℝ≥0} (ha : a ≠ 0) {C E : ℝ≥0∞} |
| (hball : ∀ x ∈ A, σ (Metric.ball x (2 * a)) ≤ C * (a : ℝ≥0∞) ^ 2) |
| (hgE : (∫⁻ x in A, g x ^ 2 ∂σ) ≤ E) |
| (hhE : (∫⁻ x in A, h x ^ 2 ∂σ) ≤ E) : |
| (∫⁻ p, codePairProximity d e A a p |
| ∂(P : Measure (BasePathCode d)).prod (Q : Measure (BasePathCode e))) ≤ C * E := by |
| change (∫⁻ p, ∫⁻ z, ballPairKernel A a z |
| ∂(baseCodeOccupation d p.1).prod (baseCodeOccupation e p.2) |
| ∂(P : Measure (BasePathCode d)).prod (Q : Measure (BasePathCode e))) ≤ C * E |
| rw [mean_code_pair_occupation d e hd he P Q _ (measurable_ballPairKernel A hA a), |
| hP, hQ, prod_withDensity hg hh, |
| lintegral_withDensity_eq_lintegral_mul _ |
| (show Measurable (fun z : Y × Y => g z.1 * h z.2) from |
| (hg.comp measurable_fst).mul (hh.comp measurable_snd)) |
| (measurable_ballPairKernel A hA a)] |
| have hb := localized_ball_pair_energy_bound σ A hA ha g h hg hh hball hgE hhE |
| simpa only [Pi.mul_apply, Function.comp_def, mul_comm] using hb |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
