File size: 2,974 Bytes
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 | 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 : @Measurable 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
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