| import OAI.Analysis.CircleDomains.Probability.CompactMeasureDensity |
| import OAI.Analysis.CircleDomains.Probability.CompactFiniteMeasureBorel |
|
|
| namespace OAI |
|
|
| /-! The exact measure-image dual conclusion for a compact family avoiding |
| zero, including support in that family and the original squared-norm bound. -/ |
| open Set Function Filter MeasureTheory |
| open scoped ENNReal NNReal |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| lemma compact_measure_image_duality |
| {Z : Type*} [TopologicalSpace Z] [PolishSpace Z] |
| [MeasurableSpace Z] [BorelSpace Z] |
| (σ : Measure Z) [IsFiniteMeasure σ] {S : Set (FiniteMeasure Z)} |
| (hS : IsCompact S) (hne : S.Nonempty) (hz : (0 : FiniteMeasure Z) ∉ S) |
| (c : ℝ≥0) (hc : 0 < c) |
| (hmod : (c : ℝ≥0∞) < measureFamilyModulus S Z σ (fun v => v.val)) : |
| ∃ (η : ProbabilityMeasure (FiniteMeasure Z)) (q : Z → ℝ≥0∞), |
| Measurable q ∧ (∀ᵐ v ∂(η : Measure (FiniteMeasure Z)), v ∈ S) ∧ |
| (∀ E, MeasurableSet E → |
| (∫⁻ v, (v : Measure Z) E ∂(η : Measure (FiniteMeasure Z))) = σ.withDensity q E) ∧ |
| (∫⁻ z, q z^2 ∂σ) ≤ (c : ℝ≥0∞)⁻¹ := by |
| let : CompactSpace S := isCompact_iff_compactSpace.mp hS |
| have hcr : (0 : ℝ) < c := hc |
| have hsqrt : 0 < Real.sqrt (c : ℝ) := Real.sqrt_pos.mpr hcr |
| have hsq : (Real.sqrt (c : ℝ))^2 = c := Real.sq_sqrt c.coe_nonneg |
| have hm : ENNReal.ofReal ((Real.sqrt (c : ℝ))^2) < |
| measureFamilyModulus S Z σ (fun v => v.val) := by simpa only [hsq,ENNReal.ofReal_coe_nnreal] using hmod |
| obtain ⟨P,q,hq,hbar,he⟩ := |
| exists_compact_measure_density_plan σ hS hne hz (Real.sqrt (c : ℝ)) hsqrt hm |
| have hval : Measurable (fun v : S => v.val) := measurable_subtype_coe |
| let η : ProbabilityMeasure (FiniteMeasure Z) := P.map (fun v : S => v.val) |
| refine ⟨η,q,hq,?_,?_,?_⟩ |
| · change ∀ᵐ v ∂(P : Measure S).map (fun v : S => v.val), v ∈ S |
| apply (ae_map_iff hval.aemeasurable (compact_finiteMeasure_measurableSet hS)).mpr |
| exact Eventually.of_forall (fun v => v.property) |
| · intro E hE |
| change (∫⁻ v, (v : Measure Z) E ∂(P : Measure S).map (fun v : S => v.val)) = _ |
| have hev : Measurable (fun v : FiniteMeasure Z => (v : Measure Z) E) := |
| (Measure.measurable_coe hE).comp measurable_subtype_coe |
| rw [lintegral_map hev hval] |
| rw [← compactMeasureBarycenter_apply P hE,hbar] |
| · refine he.trans_eq ?_ |
| rw [inv_pow,hsq,ENNReal.ofReal_inv_of_pos hcr,ENNReal.ofReal_coe_nnreal] |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
