| import OAI.Analysis.FiniteFactor.OrbitSampling.WeakLimit |
| import Mathlib.MeasureTheory.Constructions.Polish.Basic |
| import Mathlib.Topology.Instances.AddCircle.Real |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
|
|
| abbrev SourceSamplingTorus := (SourceCoordinate → UnitAddCircle) × (SourceCoordinate → UnitAddCircle) |
| |
| instance sourceSamplingTorus_compact : CompactSpace SourceSamplingTorus := inferInstance |
| |
| instance sourceSamplingTorus_borel : BorelSpace SourceSamplingTorus := Prod.borelSpace |
| |
| def sourceSamplingTorusMap (p : SourceSamplingPair) : SourceSamplingTorus := |
| (fun i => (p.1 i : UnitAddCircle),fun i => (p.2 i : UnitAddCircle)) |
| |
| theorem sourceSamplingTorusMap_continuous : Continuous sourceSamplingTorusMap := by |
| exact (continuous_pi (fun i => (AddCircle.continuous_mk' (1 : ℝ)).comp |
| ((continuous_apply i).comp continuous_fst))).prodMk |
| (continuous_pi (fun i => (AddCircle.continuous_mk' (1 : ℝ)).comp |
| ((continuous_apply i).comp continuous_snd))) |
| |
| def sourceSamplingUntrimmedTorus (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) : FiniteMeasure SourceSamplingTorus := |
| (sourceSamplingUntrimmedFinite I s θ).map sourceSamplingTorusMap |
| |
| def sourceSamplingLambdaTorus (I : Finset SourceCoordinate) : FiniteMeasure SourceSamplingTorus := |
| (sourceSamplingLambdaFinite I).map sourceSamplingTorusMap |
| |
| theorem sourceSamplingUntrimmed_torus_event : ∀ᵐ θ ∂sourceAngleLaw, |
| ∀ I : Finset SourceCoordinate, |
| Tendsto (fun s => sourceSamplingUntrimmedTorus I s θ) atTop |
| (𝓝 (sourceSamplingLambdaTorus I)) ∧ |
| ∀ f : C(SourceSamplingTorus,ℝ), |
| Tendsto (fun s => ∫ p, f p ∂(sourceSamplingUntrimmedTorus I s θ : Measure SourceSamplingTorus)) |
| atTop (𝓝 (∫ p, f p ∂(sourceSamplingLambdaTorus I : Measure SourceSamplingTorus))) := by |
| filter_upwards [sourceSamplingUntrimmed_weak_event] with θ hθ |
| intro I |
| have h := FiniteMeasure.tendsto_map_of_tendsto_of_continuous _ _ (hθ I) |
| sourceSamplingTorusMap_continuous |
| refine ⟨h,fun f => ?_⟩ |
| exact (FiniteMeasure.tendsto_iff_forall_integral_tendsto.mp h) |
| (BoundedContinuousFunction.mkOfCompact f) |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
