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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 | 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
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