| 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 |
|
|
