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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 | import OAI.Analysis.CircleDomains.Topology.NestedCompactTestPlan
import OAI.Analysis.CircleDomains.Probability.CompactPlanTestDensity
import OAI.Analysis.CircleDomains.Sobolev.MeasurePlanWeakDuality
namespace OAI
/-! Continuity of pointwise modulus on decreasing compact measure families
avoiding zero. The limiting probability plan supplies the lower bound. -/
open Set Function Filter MeasureTheory
open scoped Topology ENNReal NNReal BoundedContinuousFunction
noncomputable section
namespace CircleDomainRigidity
lemma compact_intersection_modulus_lower
{Z : Type*} [TopologicalSpace Z] [PolishSpace Z]
[MeasurableSpace Z] [BorelSpace Z]
(σ : Measure Z) [IsFiniteMeasure σ]
(T : ℕ → Set (FiniteMeasure Z)) (hT : ∀ n, IsCompact (T n))
(hne : ∀ n, (T n).Nonempty) (hanti : Antitone T)
(hz : (0 : FiniteMeasure Z) ∉ T 0) {c : ℝ≥0} (hc : 0 < c)
(hm : ∀ n, (c : ℝ≥0∞) < measureFamilyModulus (T n) Z σ (fun v => v.val)) :
(c : ℝ≥0∞) ≤ measureFamilyModulus (⋂ n, T n) Z σ (fun v => v.val) := by
let : CompactSpace (T 0) := isCompact_iff_compactSpace.mp (hT 0)
have hcr : (0 : ℝ) < c := hc
have hsq : Real.sqrt (c : ℝ)^2 = c := Real.sq_sqrt c.coe_nonneg
obtain ⟨P,hPs,hPt⟩ := exists_nested_compact_test_plan σ T hT hne hanti hz
(Real.sqrt_pos.mpr hcr) (by simpa only [hsq,ENNReal.ofReal_coe_nnreal] using hm)
obtain ⟨q,hq,hbar,he⟩ := compact_plan_density_of_test_bound σ (hT 0) P
(Real.sqrt_pos.mpr hcr) hPt
have he' : (∫⁻ z, q z^2 ∂σ) ≤ (c : ℝ≥0∞)⁻¹ := by
convert he using 1
rw [inv_pow,hsq,ENNReal.ofReal_inv_of_pos hcr,ENNReal.ofReal_coe_nnreal]
apply measureFamilyModulus_lower_of_barycenter σ (fun v : (⋂ n, T n) => v.val) P _ hq hbar hc he'
filter_upwards [hPs] with v hv
exact ⟨⟨v.val,hv⟩,rfl⟩
end CircleDomainRigidity
end
end OAI
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