openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Probability /CompactFiniteMeasureMeasurability.lean
| import OAI.Analysis.CircleDomains.Sobolev.WeakProbabilityMeasurability | |
| import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | |
| namespace OAI | |
| /-! Only the measurable inclusion of a compact weak family is needed for | |
| Riesz representation. No replacement of the evaluation sigma algebra is made. -/ | |
| open Set Function Filter MeasureTheory TopologicalSpace | |
| open scoped Topology ENNReal NNReal BoundedContinuousFunction | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma measurable_measure_of_continuous_finite | |
| {X Z : Type*} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X] | |
| [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z] [HasOuterApproxClosed Z] | |
| {F : X → FiniteMeasure Z} (hF : Continuous F) : | |
| Measurable (fun x => (F x : Measure Z)) := by | |
| apply Measurable.measure_of_isPiSystem | |
| (BorelSpace.measurable_eq.trans borel_eq_generateFrom_isClosed) isPiSystem_isClosed | |
| · intro S hS | |
| apply measurable_of_tendsto_metrizable | |
| (fun n => ((FiniteMeasure.continuous_lintegral_boundedContinuousFunction | |
| (hS.apprSeq n)).comp hF).measurable) | |
| apply tendsto_pi_nhds.mpr | |
| intro x | |
| exact HasOuterApproxClosed.tendsto_lintegral_apprSeq hS (F x : Measure Z) | |
| · simpa only [← FiniteMeasure.ennreal_mass,Function.comp_def] using | |
| (ENNReal.continuous_coe.comp (FiniteMeasure.continuous_mass.comp hF)).measurable | |
| lemma compact_finiteMeasure_measurable_le_borel | |
| {Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z] | |
| [HasOuterApproxClosed Z] (S : Set (FiniteMeasure Z)) : | |
| (inferInstance : MeasurableSpace S) ≤ borel S := by | |
| have h : S (Measure Z) (borel S) inferInstance | |
| (fun v => (v.val : Measure Z)) := by | |
| let : MeasurableSpace S := borel S | |
| let : BorelSpace S := ⟨rfl⟩ | |
| exact measurable_measure_of_continuous_finite continuous_subtype_val | |
| change ((inferInstance : MeasurableSpace (Measure Z)).comap | |
| (fun v : FiniteMeasure Z => (v : Measure Z))).comap | |
| (fun v : S => v.val) ≤ borel S | |
| simpa only [MeasurableSpace.comap_comp,Function.comp_def] using h.comap_le | |
| lemma measurable_finiteMeasure_integral | |
| {Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z] | |
| (f : Z →ᵇ ℝ) : Measurable (fun v : FiniteMeasure Z => ∫ z, f z ∂(v : Measure Z)) := by | |
| have hv : Measurable (fun v : FiniteMeasure Z => (v : Measure Z)) := measurable_subtype_coe | |
| have hp := ((Measure.measurable_lintegral f.nnrealPart.measurable_coe_ennreal_comp).comp | |
| hv).ennreal_toReal | |
| have hn := ((Measure.measurable_lintegral (-f).nnrealPart.measurable_coe_ennreal_comp).comp | |
| hv).ennreal_toReal | |
| convert hp.sub hn using 1 | |
| funext v | |
| rw [BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub] | |
| simp only [Pi.sub_apply,Function.comp_def] | |
| rw [BoundedContinuousFunction.toReal_lintegral_coe_eq_integral, | |
| BoundedContinuousFunction.toReal_lintegral_coe_eq_integral] | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |