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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 | import OAI.Analysis.CircleDomains.Sobolev.CompactMeasureL2Plan
import OAI.Analysis.CircleDomains.Probability.CompactMeasureBarycenter
import OAI.Analysis.CircleDomains.Sobolev.L2PositiveTests
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
namespace OAI
/-! The compact-family plan has the actual nonnegative Borel L² density
of its barycenter. The identity is proved on Borel sets, not just tests. -/
open Set Filter MeasureTheory
open scoped BoundedContinuousFunction ENNReal NNReal
noncomputable section
namespace CircleDomainRigidity
local instance : Fact ((1 : ℝ≥0∞) ≤ 2) := ⟨by norm_num⟩
lemma L2_ofReal_energy_eq
{Z : Type*} [MeasurableSpace Z] (σ : Measure Z)
(g : Lp ℝ 2 σ) (hg : ∀ᵐ z ∂σ, 0 ≤ g z) :
(∫⁻ z, (ENNReal.ofReal (g z))^2 ∂σ) = ENNReal.ofReal (‖g‖^2) := by
calc
(∫⁻ z, (ENNReal.ofReal (g z))^2 ∂σ) =
∫⁻ z, ENNReal.ofReal ((g z)^2) ∂σ := by
apply lintegral_congr_ae
filter_upwards [hg] with z hz
exact (ENNReal.ofReal_pow hz 2).symm
_ = ENNReal.ofReal (∫ z, (g z)^2 ∂σ) :=
(ofReal_integral_eq_lintegral_ofReal (Lp.memLp g).integrable_sq
(Eventually.of_forall (fun z => sq_nonneg (g z)))).symm
_ = ENNReal.ofReal (‖g‖^2) := by
congr 1
rw [← real_inner_self_eq_norm_sq g, L2.inner_def]
apply integral_congr_ae
exact Eventually.of_forall (fun z => by simp [pow_two])
lemma compactMeasureBarycenter_eq_withDensity
{Z : Type*} [TopologicalSpace Z] [PolishSpace Z]
[MeasurableSpace Z] [BorelSpace Z]
(σ : Measure Z) [IsFiniteMeasure σ] {S : Set (FiniteMeasure Z)}
(hS : IsCompact S) [CompactSpace S] (P : ProbabilityMeasure S)
(g : Lp ℝ 2 σ)
(hrep : ∀ f : Z →ᵇ ℝ,
(∫ v : S, (∫ z, f z ∂(v.val : Measure Z)) ∂(P : Measure S)) =
∫ z, g z * f z ∂σ) :
(∀ᵐ z ∂σ, 0 ≤ g z) ∧
compactMeasureBarycenter P = σ.withDensity (fun z => ENNReal.ofReal (g z)) := by
have hg : ∀ᵐ z ∂σ, 0 ≤ g z := by
apply L2_nonneg_of_boundedContinuous_tests σ g
intro f hf
rw [← hrep f]
exact integral_nonneg (fun v => integral_nonneg hf)
refine ⟨hg,?_⟩
let : IsFiniteMeasure (compactMeasureBarycenter P) := compactMeasureBarycenter_finite hS P
have hgi : Integrable g σ := (Lp.memLp g).integrable (by norm_num)
let : IsFiniteMeasure (σ.withDensity (fun z => ENNReal.ofReal (g z))) :=
isFiniteMeasure_withDensity ((lintegral_ofReal_ne_top_iff_integrable
(Lp.aestronglyMeasurable g) hg).mpr hgi)
apply ext_of_forall_lintegral_eq_of_IsFiniteMeasure
intro f
apply (ENNReal.toReal_eq_toReal_iff'
(BoundedContinuousFunction.lintegral_lt_top_of_nnreal (compactMeasureBarycenter P) f).ne
(BoundedContinuousFunction.lintegral_lt_top_of_nnreal
(σ.withDensity (fun z => ENNReal.ofReal (g z))) f).ne).mp
rw [BoundedContinuousFunction.toReal_lintegral_coe_eq_integral,
BoundedContinuousFunction.toReal_lintegral_coe_eq_integral]
let fR : Z →ᵇ ℝ := ⟨⟨fun z => (f z : ℝ), NNReal.continuous_coe.comp f.continuous⟩,
f.map_bounded'⟩
change (∫ z, fR z ∂compactMeasureBarycenter P) =
∫ z, fR z ∂σ.withDensity (fun z => ENNReal.ofReal (g z))
rw [compactMeasureBarycenter_integral_nonneg P fR (fun z => (f z).coe_nonneg), hrep fR,
integral_withDensity_eq_integral_toReal_smul (Lp.stronglyMeasurable g).measurable.ennreal_ofReal
(Eventually.of_forall (fun z => ENNReal.ofReal_lt_top))]
apply integral_congr_ae
filter_upwards [hg] with z hz
simp only [ENNReal.toReal_ofReal hz,smul_eq_mul]
lemma exists_compact_measure_density_plan
{Z : Type*} [TopologicalSpace Z] [PolishSpace Z]
[MeasurableSpace Z] [BorelSpace Z]
(σ : Measure Z) [IsFiniteMeasure σ] {S : Set (FiniteMeasure 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) (q : Z → ℝ≥0∞), Measurable q ∧
compactMeasureBarycenter P = σ.withDensity q ∧
(∫⁻ z, q z^2 ∂σ) ≤ ENNReal.ofReal (c⁻¹^2) := by
obtain ⟨P,g,hgn,hrep⟩ := exists_compact_measure_L2_test_plan σ hS hne hz c hc hmod
obtain ⟨hg,hbar⟩ := compactMeasureBarycenter_eq_withDensity σ hS P g hrep
refine ⟨P,fun z => ENNReal.ofReal (g z), (Lp.stronglyMeasurable g).measurable.ennreal_ofReal,
hbar,?_⟩
rw [L2_ofReal_energy_eq σ g hg]
apply ENNReal.ofReal_le_ofReal
exact pow_le_pow_left₀ (norm_nonneg g) hgn 2
end CircleDomainRigidity
end
end OAI
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