openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /CompactMeasureMajorantCost.lean
| import OAI.Analysis.CircleDomains.Probability.CompactMeasureMajorants | |
| import Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions | |
| import Mathlib.Topology.ContinuousMap.CompactlySupported | |
| namespace OAI | |
| /-! The actual least L² cost of a bounded continuous majorant on a compact | |
| family of finite measures. Feasibility follows from its positive minimum | |
| mass; sublinearity follows from the triangle inequality in L². -/ | |
| open Set 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)) | |
| private def majorantNorms (u : C_c(S,ℝ)) : Set ℝ := | |
| {r | ∃ f : Z →ᵇ ℝ, (∀ v : S, u v ≤ compactMeasureTestOperator S f v) ∧ | |
| ‖BoundedContinuousFunction.toLp 2 σ ℝ f‖ = r} | |
| def compactMeasureMajorantCost (u : C_c(S,ℝ)) : ℝ := sInf (majorantNorms σ S u) | |
| private lemma majorantNorms_bddBelow (u : C_c(S,ℝ)) : BddBelow (majorantNorms σ S u) := | |
| ⟨0,fun _ ⟨_,_,he⟩ => he ▸ norm_nonneg _⟩ | |
| variable {S} | |
| private lemma majorantNorms_nonempty (hS : IsCompact S) (hne : S.Nonempty) | |
| (hz : (0 : FiniteMeasure Z) ∉ S) [CompactSpace S] (u : C_c(S,ℝ)) : | |
| (majorantNorms σ S u).Nonempty := by | |
| obtain ⟨r,_,hr⟩ := exists_compact_measure_constant_majorant hS hne hz u.toContinuousMap | |
| exact ⟨_,BoundedContinuousFunction.const Z r,hr,rfl⟩ | |
| lemma compactMeasureMajorantCost_nonneg (hS : IsCompact S) (hne : S.Nonempty) | |
| (hz : (0 : FiniteMeasure Z) ∉ S) [CompactSpace S] (u : C_c(S,ℝ)) : | |
| 0 ≤ compactMeasureMajorantCost σ S u := | |
| le_csInf (majorantNorms_nonempty σ hS hne hz u) | |
| (fun _ ⟨_,_,he⟩ => he ▸ norm_nonneg _) | |
| lemma le_compactMeasureMajorantCost (hS : IsCompact S) (hne : S.Nonempty) | |
| (hz : (0 : FiniteMeasure Z) ∉ S) [CompactSpace S] | |
| (u : C_c(S,ℝ)) {a : ℝ} | |
| (ha : ∀ f : Z →ᵇ ℝ, (∀ v : S, u v ≤ compactMeasureTestOperator S f v) → | |
| a ≤ ‖BoundedContinuousFunction.toLp 2 σ ℝ f‖) : | |
| a ≤ compactMeasureMajorantCost σ S u := by | |
| apply le_csInf (majorantNorms_nonempty σ hS hne hz u) | |
| rintro _ ⟨f,hf,rfl⟩ | |
| exact ha f hf | |
| lemma compactMeasureMajorantCost_le (u : C_c(S,ℝ)) (f : Z →ᵇ ℝ) | |
| (hf : ∀ v : S, u v ≤ compactMeasureTestOperator S f v) : | |
| compactMeasureMajorantCost σ S u ≤ ‖BoundedContinuousFunction.toLp 2 σ ℝ f‖ := | |
| csInf_le (majorantNorms_bddBelow σ S u) ⟨f,hf,rfl⟩ | |
| lemma compactMeasureMajorantCost_nonpos (hS : IsCompact S) (hne : S.Nonempty) | |
| (hz : (0 : FiniteMeasure Z) ∉ S) [CompactSpace S] | |
| (u : C_c(S,ℝ)) (hu : ∀ v, u v ≤ 0) : compactMeasureMajorantCost σ S u = 0 := by | |
| apply le_antisymm _ (compactMeasureMajorantCost_nonneg σ hS hne hz u) | |
| have hb := compactMeasureMajorantCost_le σ u (0 : Z →ᵇ ℝ) | |
| (fun v => by simpa using hu v) | |
| simpa using hb | |
| lemma compactMeasureMajorantCost_smul_le (hS : IsCompact S) (hne : S.Nonempty) | |
| (hz : (0 : FiniteMeasure Z) ∉ S) [CompactSpace S] | |
| (r : ℝ) (hr : 0 < r) (u : C_c(S,ℝ)) : | |
| compactMeasureMajorantCost σ S (r • u) ≤ r * compactMeasureMajorantCost σ S u := by | |
| have hdiv : compactMeasureMajorantCost σ S (r • u) / r ≤ | |
| compactMeasureMajorantCost σ S u := by | |
| apply le_csInf (majorantNorms_nonempty σ hS hne hz u) | |
| rintro _ ⟨f,hf,rfl⟩ | |
| apply (div_le_iff₀ hr).mpr | |
| have hb := compactMeasureMajorantCost_le σ (r • u) (r • f) (fun v => by | |
| simpa using mul_le_mul_of_nonneg_left (hf v) hr.le) | |
| simpa only [map_smul,norm_smul,Real.norm_eq_abs,abs_of_pos hr,smul_eq_mul,mul_comm] using hb | |
| have hb := (div_le_iff₀ hr).mp hdiv | |
| simpa only [mul_comm] using hb | |
| lemma compactMeasureMajorantCost_smul (hS : IsCompact S) (hne : S.Nonempty) | |
| (hz : (0 : FiniteMeasure Z) ∉ S) [CompactSpace S] | |
| (r : ℝ) (hr : 0 < r) (u : C_c(S,ℝ)) : | |
| compactMeasureMajorantCost σ S (r • u) = r * compactMeasureMajorantCost σ S u := by | |
| apply le_antisymm (compactMeasureMajorantCost_smul_le σ hS hne hz r hr u) | |
| have hb := compactMeasureMajorantCost_smul_le σ hS hne hz r⁻¹ (inv_pos.mpr hr) (r • u) | |
| rw [inv_smul_smul₀ hr.ne'] at hb | |
| have hc := mul_le_mul_of_nonneg_left hb hr.le | |
| simpa only [← mul_assoc,mul_inv_cancel₀ hr.ne',one_mul] using hc | |
| lemma compactMeasureMajorantCost_add (hS : IsCompact S) (hne : S.Nonempty) | |
| (hz : (0 : FiniteMeasure Z) ∉ S) [CompactSpace S] (u v : C_c(S,ℝ)) : | |
| compactMeasureMajorantCost σ S (u+v) ≤ | |
| compactMeasureMajorantCost σ S u + compactMeasureMajorantCost σ S v := by | |
| have hb (f g : Z →ᵇ ℝ) | |
| (hf : ∀ a : S, u a ≤ compactMeasureTestOperator S f a) | |
| (hg : ∀ a : S, v a ≤ compactMeasureTestOperator S g a) : | |
| compactMeasureMajorantCost σ S (u+v) ≤ | |
| ‖BoundedContinuousFunction.toLp 2 σ ℝ f‖ + ‖BoundedContinuousFunction.toLp 2 σ ℝ g‖ := by | |
| have hfg := compactMeasureMajorantCost_le σ (u+v) (f+g) (fun a => by | |
| simpa using add_le_add (hf a) (hg a)) | |
| have hn := norm_add_le (BoundedContinuousFunction.toLp 2 σ ℝ f) | |
| (BoundedContinuousFunction.toLp 2 σ ℝ g) | |
| exact hfg.trans (by simpa only [map_add] using hn) | |
| have hu (f : Z →ᵇ ℝ) (hf : ∀ a : S, u a ≤ compactMeasureTestOperator S f a) : | |
| compactMeasureMajorantCost σ S (u+v) - ‖BoundedContinuousFunction.toLp 2 σ ℝ f‖ ≤ | |
| compactMeasureMajorantCost σ S v := by | |
| apply le_csInf (majorantNorms_nonempty σ hS hne hz v) | |
| rintro _ ⟨g,hg,rfl⟩ | |
| linarith [hb f g hf hg] | |
| have hv : compactMeasureMajorantCost σ S (u+v) - compactMeasureMajorantCost σ S v ≤ | |
| compactMeasureMajorantCost σ S u := by | |
| apply le_csInf (majorantNorms_nonempty σ hS hne hz u) | |
| rintro _ ⟨f,hf,rfl⟩ | |
| linarith [hu f hf] | |
| linarith | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |