File size: 4,492 Bytes
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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.Analysis.CircleDomains.Probability.CompactMeasureContinuity
import OAI.Analysis.CircleDomains.Modulus.ModulusFiniteMesh
namespace OAI
/-! A continuous presentation inside a compact family avoiding zero has a
compact subfamily retaining every prescribed strict modulus threshold. -/
open Set Function Filter MeasureTheory TopologicalSpace Metric
open scoped Topology ENNReal NNReal
noncomputable section
namespace CircleDomainRigidity
lemma measureFamilyModulus_zero_of_isEmpty
{A Z : Type*} [IsEmpty A] [MeasurableSpace Z]
(σ : Measure Z) (K : A → FiniteMeasure Z) : measureFamilyModulus A Z σ K = 0 := by
apply le_antisymm _ bot_le
have h := measureFamilyModulus_le_admissible σ K
(h := fun _ => 0) measurable_const (fun a => isEmptyElim a)
change measureFamilyModulus A Z σ K ≤ 0
simpa only [zero_pow (by norm_num : (2:ℕ) ≠ 0),lintegral_zero] using h
lemma exists_compact_subfamily_of_continuous_image
{X Z : Type*} [MetricSpace X] [CompleteSpace X] [SeparableSpace X] [Nonempty X]
[TopologicalSpace Z] [PolishSpace Z] [MeasurableSpace Z] [BorelSpace Z]
(σ : Measure Z) [IsFiniteMeasure σ] {T : Set (FiniteMeasure Z)}
(hT : IsCompact T) (hz : (0 : FiniteMeasure Z) ∉ T)
(f : X → T) (hf : Continuous f) {c : ℝ≥0∞}
(hc : c < measureFamilyModulus X Z σ (fun x => (f x).val)) :
∃ S : Set (FiniteMeasure Z), IsCompact S ∧ S ⊆ range (fun x => (f x).val) ∧
c < measureFamilyModulus S Z σ (fun v => v.val) := by
classical
let : CompactSpace T := isCompact_iff_compactSpace.mp hT
let : PolishSpace T := compact_finiteMeasure_polish hT
let : MetricSpace T := metrizableSpaceMetric T
obtain ⟨d,hcd,hd⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hc
have hd0 : 0 < d := by
have hp : (0 : ℝ≥0∞) < d := (show (0 : ℝ≥0∞) ≤ c from zero_le).trans_lt hcd
exact_mod_cast hp
have heq : measureFamilyModulus (univ : Set X) Z σ (fun x => (f x.val).val) =
measureFamilyModulus X Z σ (fun x => (f x).val) := by
apply measureFamilyModulus_eq_of_range_eq
ext v
constructor
· rintro ⟨x,rfl⟩
exact ⟨x.val,rfl⟩
· rintro ⟨x,rfl⟩
exact ⟨⟨x,mem_univ x⟩,rfl⟩
obtain ⟨C,F,_,hC,hanti,hmesh,hlarge⟩ := exists_modulus_finite_mesh σ
(fun x => (f x).val) isClosed_univ (c := (d : ℝ≥0∞)) (heq.symm ▸ hd)
have hne (n : ℕ) : (C n).Nonempty := by
by_contra hn
let : IsEmpty (C n) := ⟨fun x => hn ⟨x.val,x.property⟩⟩
have hm := hlarge n
rw [measureFamilyModulus_zero_of_isEmpty] at hm
exact (not_lt_of_ge bot_le) hm
let U (n : ℕ) : Set (FiniteMeasure Z) := Subtype.val '' closure (f '' C n)
have hUc (n : ℕ) : IsCompact (U n) := isClosed_closure.isCompact.image continuous_subtype_val
have hUn (n : ℕ) : (U n).Nonempty := by
obtain ⟨x,hx⟩ := hne n
exact ⟨(f x).val,⟨f x,subset_closure (mem_image_of_mem f hx),rfl⟩⟩
have hUa : Antitone U := fun n m hnm => image_mono (closure_mono (image_mono (hanti hnm)))
have hUz : (0 : FiniteMeasure Z) ∉ U 0 := by
rintro ⟨w,_,hw⟩
exact hz (hw ▸ w.property)
have hUl (n : ℕ) : (d : ℝ≥0∞) < measureFamilyModulus (U n) Z σ (fun v => v.val) := by
apply (hlarge n).trans_le
apply measureFamilyModulus_mono_range
rintro _ ⟨x,rfl⟩
exact ⟨⟨(f x.val).val,⟨f x.val,subset_closure (mem_image_of_mem f x.property),rfl⟩⟩,rfl⟩
have hlow := compact_intersection_modulus_lower σ U hUc hUn hUa hUz hd0 hUl
let S : Set (FiniteMeasure Z) := (fun x => (f x).val) '' (⋂ n, C n)
have hSc : IsCompact S := (compact_iInter_of_finite_mesh C hC F hmesh).image
(continuous_subtype_val.comp hf)
have hIS : (⋂ n, U n) ⊆ S := by
intro v hv
obtain ⟨w,_,hwv⟩ := mem_iInter.mp hv 0
have hw : w ∈ ⋂ n, closure (f '' C n) := by
apply mem_iInter.mpr
intro n
obtain ⟨w',hw',he⟩ := mem_iInter.mp hv n
have heq : w' = w := Subtype.ext (he.trans hwv.symm)
exact heq ▸ hw'
rw [← image_iInter_eq_iInter_closure_of_finite_mesh C hC hanti F hmesh f hf] at hw
obtain ⟨x,hx,hfx⟩ := hw
exact ⟨x,hx,(congrArg Subtype.val hfx).trans hwv⟩
refine ⟨S,hSc,?_ ,hcd.trans_le (hlow.trans ?_)⟩
· rintro v ⟨x,_,rfl⟩
exact mem_range_self x
· apply measureFamilyModulus_mono_range
rintro _ ⟨v,rfl⟩
exact ⟨⟨v.val,hIS v.property⟩,rfl⟩
end CircleDomainRigidity
end
end OAI
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