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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 | 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
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