openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Modulus /CompactContinuousImageModulus.lean
| 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 | |