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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 54 55 56 57 58 | import OAI.Analysis.CircleDomains.Modulus.CompactContinuousImageModulus
import OAI.Analysis.CircleDomains.Probability.PositiveMassCompactLocalization
import Mathlib.MeasureTheory.Constructions.Polish.Basic
namespace OAI
/-! Compact inner approximation for the pointwise modulus of a Borel image.
The compatible Polish topology is refined only after compact localization. -/
open Set Function Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
noncomputable section
namespace CircleDomainRigidity
lemma exists_compact_subfamily_of_borel_image
{A Z : Type*} [MeasurableSpace A] [StandardBorelSpace A]
[TopologicalSpace Z] [PolishSpace Z] [MeasurableSpace Z] [BorelSpace Z]
(σ : Measure Z) [IsFiniteMeasure σ] (K : A → FiniteMeasure Z) (hKm : Measurable K)
(hK : ∀ a, K a ≠ 0) {c : ℝ≥0∞}
(hc : c < measureFamilyModulus A Z σ K) :
∃ S : Set (FiniteMeasure Z), IsCompact S ∧ S ⊆ range K ∧
c < measureFamilyModulus S Z σ (fun v => v.val) := by
classical
obtain ⟨T,hT,hTz,hmod⟩ := exists_compact_nonzero_modulus_localization σ K hK hc
let B : Set A := {a | K a ∈ T}
have hBm : MeasurableSet B := hKm (compact_finiteMeasure_measurableSet hT)
let : StandardBorelSpace B := hBm.standardBorel
let := upgradeStandardBorel B
let : BorelSpace T := compact_finiteMeasure_borel hT
let : PolishSpace T := compact_finiteMeasure_polish hT
let f : B → T := fun a => ⟨K a.val,a.property⟩
have hfm : Measurable f := (hKm.comp measurable_subtype_coe).subtype_mk
obtain ⟨t,hle,hfc,ht⟩ := hfm.exists_continuous
have hbe : @BorelSpace B t inferInstance := by
constructor
rw [eq_borel_upgradeStandardBorel B,borel_eq_borel_of_le ht _ hle]
infer_instance
let : TopologicalSpace B := t
let : PolishSpace B := ht
let : BorelSpace B := hbe
let := upgradeIsCompletelyMetrizable B
cases isEmpty_or_nonempty B with
| inl hB =>
have hz := measureFamilyModulus_zero_of_isEmpty σ (fun a : B => K a.val)
rw [hz] at hmod
exact (not_lt_of_ge bot_le hmod).elim
| inr hB =>
obtain ⟨S,hS,hSf,hSc⟩ := exists_compact_subfamily_of_continuous_image σ hT hTz f hfc hmod
refine ⟨S,hS,?_,hSc⟩
rintro v hv
obtain ⟨a,ha⟩ := hSf hv
exact ⟨a.val,ha⟩
end CircleDomainRigidity
end
end OAI
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