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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 | import Mathlib.MeasureTheory.Measure.Portmanteau
import Mathlib.Topology.Semicontinuity.Basic
namespace OAI
/-! Support in a closed set is retained by weak limits of probability laws. -/
open Set Filter MeasureTheory
open scoped Topology ENNReal
noncomputable section
namespace CircleDomainRigidity
lemma isClosed_probability_supported_closed
{X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
[HasOuterApproxClosed X] {F : Set X} (hF : IsClosed F) :
IsClosed {P : ProbabilityMeasure X | ∀ᵐ x ∂(P : Measure X), x ∈ F} := by
have hsc : UpperSemicontinuous (fun P : ProbabilityMeasure X => (P : Measure X) F) :=
upperSemicontinuous_iff_limsup_le.mpr (fun P =>
ProbabilityMeasure.limsup_measure_closed_le_of_tendsto tendsto_id hF)
have he : {P : ProbabilityMeasure X | ∀ᵐ x ∂(P : Measure X), x ∈ F} =
{P : ProbabilityMeasure X | (1 : ℝ≥0∞) ≤ (P : Measure X) F} := by
ext P
change (∀ᵐ x ∂(P : Measure X), x ∈ F) ↔ (1 : ℝ≥0∞) ≤ (P : Measure X) F
constructor
· intro h
have heq : F =ᵐ[(P : Measure X)] univ := h.mono (fun x hx => by simp [hx])
rw [measure_congr heq,measure_univ]
· intro h
have hm : (P : Measure X) F = 1 := le_antisymm ((measure_mono (subset_univ F)).trans_eq measure_univ) h
have hc : (P : Measure X) Fᶜ = 0 := by
rw [measure_compl hF.measurableSet (measure_ne_top _ _),hm,measure_univ,tsub_self]
apply ae_iff.mpr
change (P : Measure X) Fᶜ = 0
exact hc
rw [he]
exact hsc.isClosed_preimage 1
lemma probability_supported_closed_of_tendsto
{X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
[HasOuterApproxClosed X] {P : ℕ → ProbabilityMeasure X} {Q : ProbabilityMeasure X}
(hlim : Tendsto P atTop (𝓝 Q)) {F : Set X} (hF : IsClosed F)
(hsupp : ∀ᶠ n in atTop, ∀ᵐ x ∂(P n : Measure X), x ∈ F) :
∀ᵐ x ∂(Q : Measure X), x ∈ F :=
(isClosed_probability_supported_closed hF).mem_of_tendsto hlim hsupp
end CircleDomainRigidity
end
end OAI
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