openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /CompactMeasureSeparation.lean
| import OAI.Analysis.CircleDomains.Probability.CompactMeasureMajorantCost | |
| import OAI.Analysis.CircleDomains.Modulus.CompactMajorantSeparation | |
| import OAI.Analysis.CircleDomains.Probability.ContinuousMeasureModulus | |
| import OAI.Analysis.CircleDomains.Probability.CompactFiniteMeasureMeasurability | |
| namespace OAI | |
| /-! Positive modulus separates the actual compact-family majorant cost | |
| from a constant functional. The separating functional is constructed. -/ | |
| open Set Filter MeasureTheory | |
| open scoped CompactlySupported BoundedContinuousFunction ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| local instance : Fact ((1 : ℝ≥0∞) ≤ 2) := ⟨by norm_num⟩ | |
| variable {Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z] | |
| (σ : Measure Z) [IsFiniteMeasure σ] {S : Set (FiniteMeasure Z)} | |
| lemma compactMeasureMajorantCost_const_lower | |
| (hS : IsCompact S) (hne : S.Nonempty) (hz : (0 : FiniteMeasure Z) ∉ S) | |
| [CompactSpace S] (c : ℝ) (hc : 0 < c) | |
| (hmod : ENNReal.ofReal (c^2) < measureFamilyModulus S Z σ (fun v => v.val)) | |
| (r : ℝ) : r*c ≤ compactMeasureMajorantCost σ S | |
| (r • (⟨1,HasCompactSupport.of_compactSpace (1 : C(S,ℝ))⟩ : C_c(S,ℝ))) := by | |
| by_cases hr : r ≤ 0 | |
| · exact (mul_nonpos_of_nonpos_of_nonneg hr hc.le).trans | |
| (compactMeasureMajorantCost_nonneg σ hS hne hz _) | |
| have hr : 0 < r := lt_of_not_ge hr | |
| apply le_compactMeasureMajorantCost σ hS hne hz | |
| intro f hf | |
| let g : Z →ᵇ ℝ := r⁻¹ • f | |
| have ha (v : S) : 1 ≤ ∫ z, g z ∂(v.val : Measure Z) := by | |
| have hv : r ≤ ∫ z, f z ∂(v.val : Measure Z) := by | |
| simpa [compactMeasureTestOperator] using hf v | |
| have hb := mul_le_mul_of_nonneg_left hv (inv_nonneg.mpr hr.le) | |
| simpa [g,BoundedContinuousFunction.smul_apply,integral_const_mul, | |
| inv_mul_cancel₀ hr.ne'] using hb | |
| have hb := measureFamilyModulus_le_boundedContinuous_norm σ (fun v : S => v.val) g ha | |
| have hsq : c^2 < ‖BoundedContinuousFunction.toLp 2 σ ℝ g‖^2 := | |
| (ENNReal.ofReal_lt_ofReal_iff'.mp (hmod.trans_le hb)).1 | |
| have hlin : c < ‖BoundedContinuousFunction.toLp 2 σ ℝ g‖ := by | |
| nlinarith [norm_nonneg (BoundedContinuousFunction.toLp 2 σ ℝ g)] | |
| have hnorm : ‖BoundedContinuousFunction.toLp 2 σ ℝ g‖ = | |
| r⁻¹ * ‖BoundedContinuousFunction.toLp 2 σ ℝ f‖ := by | |
| simp only [g,map_smul,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hr)] | |
| rw [hnorm] at hlin | |
| have hmul := mul_lt_mul_of_pos_left hlin hr | |
| have hmul' : r*c < ‖BoundedContinuousFunction.toLp 2 σ ℝ f‖ := by | |
| simpa only [← mul_assoc,mul_inv_cancel₀ hr.ne',one_mul] using hmul | |
| exact hmul'.le | |
| lemma exists_compact_measure_separating_functional | |
| (hS : IsCompact S) (hne : S.Nonempty) (hz : (0 : FiniteMeasure Z) ∉ S) | |
| [CompactSpace S] (c : ℝ) (hc : 0 < c) | |
| (hmod : ENNReal.ofReal (c^2) < measureFamilyModulus S Z σ (fun v => v.val)) : | |
| ∃ Λ : C_c(S,ℝ) →ₚ[ℝ] ℝ, | |
| Λ ⟨1,HasCompactSupport.of_compactSpace (1 : C(S,ℝ))⟩ = c ∧ | |
| ∀ u, Λ u ≤ compactMeasureMajorantCost σ S u := by | |
| let : Nonempty S := hne.to_subtype | |
| apply exists_compact_positive_functional_of_majorant_cost | |
| (compactMeasureMajorantCost σ S) | |
| · exact fun r hr u => compactMeasureMajorantCost_smul σ hS hne hz r hr u | |
| · exact compactMeasureMajorantCost_add σ hS hne hz | |
| · intro u hu | |
| exact compactMeasureMajorantCost_nonpos σ hS hne hz (-u) (fun v => by | |
| simpa using neg_nonpos.mpr (hu v)) | |
| · exact compactMeasureMajorantCost_const_lower σ hS hne hz c hc hmod | |
| lemma exists_compact_measure_test_plan [PolishSpace Z] | |
| (hS : IsCompact S) (hne : S.Nonempty) (hz : (0 : FiniteMeasure Z) ∉ S) | |
| [CompactSpace S] (c : ℝ) (hc : 0 < c) | |
| (hmod : ENNReal.ofReal (c^2) < measureFamilyModulus S Z σ (fun v => v.val)) : | |
| ∃ P : ProbabilityMeasure S, ∀ f : Z →ᵇ ℝ, | |
| c * (∫ v : S, (∫ z, f z ∂(v.val : Measure Z)) ∂(P : Measure S)) ≤ | |
| ‖BoundedContinuousFunction.toLp 2 σ ℝ f‖ := by | |
| obtain ⟨Λ,hΛ1,hΛ⟩ := exists_compact_measure_separating_functional σ hS hne hz c hc hmod | |
| let L : C_c(S,ℝ) →ₚ[ℝ] ℝ := PositiveLinearMap.mk₀ (c⁻¹ • Λ.toLinearMap) | |
| (fun u hu => mul_nonneg (inv_nonneg.mpr hc.le) (Λ.map_nonneg hu)) | |
| have hL1 : L ⟨1,HasCompactSupport.of_compactSpace (1 : C(S,ℝ))⟩ = 1 := by | |
| change c⁻¹ * Λ _ = 1 | |
| rw [hΛ1,inv_mul_cancel₀ hc.ne'] | |
| obtain ⟨P,hP⟩ := exists_probability_of_compact_positive_functional_trim | |
| (compact_finiteMeasure_measurable_le_borel S) L hL1 | |
| refine ⟨P,?_⟩ | |
| intro f | |
| let u : C_c(S,ℝ) := ⟨compactMeasureTestOperator S f, | |
| HasCompactSupport.of_compactSpace (compactMeasureTestOperator S f)⟩ | |
| have hi : c * (∫ v : S, u v ∂(P : Measure S)) = Λ u := by | |
| rw [hP u ((measurable_finiteMeasure_integral f).comp measurable_subtype_coe)] | |
| change c * (c⁻¹ * Λ u) = Λ u | |
| rw [← mul_assoc,mul_inv_cancel₀ hc.ne',one_mul] | |
| have hb := (hΛ u).trans (compactMeasureMajorantCost_le σ u f (fun _ => le_rfl)) | |
| rw [← hi] at hb | |
| exact hb | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |