| import OAI.Analysis.CircleDomains.Probability.PublishedMeasureDuality |
| import Mathlib.Topology.Order.Compact |
| import Mathlib.Topology.ContinuousMap.Compact |
| import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction |
|
|
| namespace OAI |
|
|
| /-! Continuous test integrals on a compact family of finite measures and |
| the uniform positive mass needed for the compact duality construction. -/ |
| open Set MeasureTheory |
| open scoped BoundedContinuousFunction ENNReal NNReal |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| variable {Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z] |
|
|
| lemma compact_family_positive_mass {S : Set (FiniteMeasure Z)} |
| (hS : IsCompact S) (hne : S.Nonempty) (hz : (0 : FiniteMeasure Z) ∉ S) : |
| ∃ m : ℝ≥0, 0 < m ∧ ∀ v ∈ S, m ≤ v.mass := by |
| obtain ⟨v,hv,hmin⟩ := hS.exists_isMinOn hne FiniteMeasure.continuous_mass.continuousOn |
| refine ⟨v.mass,?_,hmin⟩ |
| exact pos_iff_ne_zero.mpr ((FiniteMeasure.mass_nonzero_iff v).mpr (fun he => hz (he ▸ hv))) |
|
|
| / |
| def compactMeasureTestOperator (S : Set (FiniteMeasure Z)) : |
| (Z →ᵇ ℝ) →ₗ[ℝ] C(S,ℝ) where |
| toFun f := ⟨fun v => ∫ z, f z ∂(v.val : Measure Z), |
| (FiniteMeasure.continuous_integral_boundedContinuousFunction f).comp continuous_subtype_val⟩ |
| map_add' f g := by |
| ext v |
| exact integral_add (f.integrable (μ := (v.val : Measure Z))) |
| (g.integrable (μ := (v.val : Measure Z))) |
| map_smul' r f := by |
| ext v |
| exact integral_smul r f |
|
|
| lemma compactMeasureTestOperator_const (S : Set (FiniteMeasure Z)) (r : ℝ) (v : S) : |
| compactMeasureTestOperator S (BoundedContinuousFunction.const Z r) v = |
| r * (v.val.mass : ℝ) := by |
| simp [compactMeasureTestOperator,integral_const,measureReal_def, |
| ← FiniteMeasure.ennreal_mass,mul_comm] |
|
|
| lemma exists_compact_measure_constant_majorant {S : Set (FiniteMeasure Z)} |
| (hS : IsCompact S) (hne : S.Nonempty) (hz : (0 : FiniteMeasure Z) ∉ S) |
| [CompactSpace S] (u : C(S,ℝ)) : |
| ∃ r : ℝ, 0 ≤ r ∧ ∀ v : S, |
| u v ≤ compactMeasureTestOperator S (BoundedContinuousFunction.const Z r) v := by |
| obtain ⟨m,hm,hmass⟩ := compact_family_positive_mass hS hne hz |
| have hm' : 0 < (m : ℝ) := hm |
| refine ⟨‖u‖ / m,div_nonneg (norm_nonneg _) hm'.le,?_⟩ |
| intro v |
| rw [compactMeasureTestOperator_const] |
| have hv : (m : ℝ) ≤ v.val.mass := hmass v.val v.property |
| calc |
| u v ≤ ‖u‖ := u.apply_le_norm v |
| _ = (‖u‖ / m) * m := (div_mul_cancel₀ _ hm'.ne').symm |
| _ ≤ (‖u‖ / m) * (v.val.mass : ℝ) := |
| mul_le_mul_of_nonneg_left hv (div_nonneg (norm_nonneg _) hm'.le) |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
