File size: 2,939 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 | 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
|