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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 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | 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
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