| import OAI.Analysis.FiniteFactor.OrbitSampling.OrderedRestriction |
| import OAI.Analysis.FiniteFactor.OrbitSampling.CompactTorus |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
|
|
| def sourceSamplingTrimmedFinite (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) : FiniteMeasure SourceSamplingPair := |
| ⟨sourceSamplingTrimmed I s θ,inferInstance⟩ |
|
|
| def sourceSamplingTrimmedTorus (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) : FiniteMeasure SourceSamplingTorus := |
| (sourceSamplingTrimmedFinite I s θ).map sourceSamplingTorusMap |
|
|
| theorem sourceSampling_discarded_tendsto (I : Finset SourceCoordinate) |
| (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) |
| (hmass : Tendsto (fun s => ((sourceSamplingUntrimmed I s θ) Set.univ).toReal) |
| atTop (𝓝 (((sourceSamplingLambda I) Set.univ).toReal))) : |
| Tendsto (fun s => ((sourceSamplingUntrimmed I s θ) sourceSamplingOrderedᶜ).toReal) |
| atTop (𝓝 0) := by |
| have hlim := hmass.mul sourceSampling_tail_error_tendsto |
| rw [mul_zero] at hlim |
| apply squeeze_zero (fun s => ENNReal.toReal_nonneg) _ hlim |
| intro s |
| have h := ENNReal.toReal_mono (by finiteness) |
| (sourceSamplingUntrimmed_discarded_le I s θ hθ) |
| simpa only [ENNReal.toReal_mul,ENNReal.toReal_ofReal (by positivity : |
| 0 ≤ 4*(sourceBlockSize s : ℝ)^(-(sourceBlockSize s : ℝ)/20))] using h |
|
|
| def finiteMeasureRestrict {Ω : Type*} [MeasurableSpace Ω] |
| (μ : FiniteMeasure Ω) (A : Set Ω) : FiniteMeasure Ω := |
| ⟨(μ : Measure Ω).restrict A,inferInstance⟩ |
|
|
| / |
| weak limit. The norm bound gives the actual error for every test function. -/ |
| theorem finiteMeasure_restrict_tendsto {Ω : Type*} [MeasurableSpace Ω] |
| [TopologicalSpace Ω] [OpensMeasurableSpace Ω] |
| (μ : ℕ → FiniteMeasure Ω) (ν : FiniteMeasure Ω) (A : Set Ω) (hA : MeasurableSet A) |
| (hμ : Tendsto μ atTop (𝓝 ν)) |
| (hloss : Tendsto (fun n => ((μ n : Measure Ω) Aᶜ).toReal) atTop (𝓝 0)) : |
| Tendsto (fun n => finiteMeasureRestrict (μ n) A) |
| atTop (𝓝 ν) := by |
| apply FiniteMeasure.tendsto_of_forall_integral_tendsto |
| intro f |
| have hf := FiniteMeasure.tendsto_iff_forall_integral_tendsto.mp hμ f |
| have he : Tendsto (fun n => ∫ x in Aᶜ, f x ∂(μ n : Measure Ω)) atTop (𝓝 0) := by |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr |
| have hb := hloss.const_mul ‖f‖ |
| rw [mul_zero] at hb |
| apply squeeze_zero (fun n => norm_nonneg _) _ hb |
| intro n |
| have hh := norm_integral_le_of_norm_le_const |
| (μ := (μ n : Measure Ω).restrict Aᶜ) |
| (Eventually.of_forall (fun x => f.norm_coe_le_norm x)) |
| simpa only [measureReal_def,Measure.restrict_apply MeasurableSet.univ,Set.univ_inter] using hh |
| have hsub := hf.sub he |
| rw [sub_zero] at hsub |
| convert hsub using 1 |
| funext n |
| exact (eq_sub_iff_add_eq.mpr (integral_add_compl (μ := (μ n : Measure Ω)) hA (f.integrable (μ := (μ n : Measure Ω))))) |
|
|
| / |
| all continuous tests, on one probability-one angle event. -/ |
| theorem sourceSamplingTrimmed_weak_event : ∀ᵐ θ ∂sourceAngleLaw, |
| ∀ I : Finset SourceCoordinate, |
| Tendsto (fun s => sourceSamplingTrimmedFinite I s θ) atTop (𝓝 (sourceSamplingLambdaFinite I)) ∧ |
| Tendsto (fun s => sourceSamplingTrimmedTorus I s θ) atTop (𝓝 (sourceSamplingLambdaTorus I)) ∧ |
| ∀ f : C(SourceSamplingTorus,ℝ), |
| Tendsto (fun s => ∫ p, f p ∂(sourceSamplingTrimmedTorus I s θ : Measure SourceSamplingTorus)) |
| atTop (𝓝 (∫ p, f p ∂(sourceSamplingLambdaTorus I : Measure SourceSamplingTorus))) := by |
| filter_upwards [sourceSamplingUntrimmed_weak_event,sourceSampling_mass_limit_event,sourceAngleLaw_admissible] |
| with θ hw hm ha |
| intro I |
| have h := finiteMeasure_restrict_tendsto (fun s => sourceSamplingUntrimmedFinite I s θ) |
| (sourceSamplingLambdaFinite I) sourceSamplingOrdered sourceSamplingOrdered_measurable |
| (hw I) (sourceSampling_discarded_tendsto I θ ha (hm I)) |
| have ht := FiniteMeasure.tendsto_map_of_tendsto_of_continuous _ _ h sourceSamplingTorusMap_continuous |
| refine ⟨h,ht,fun f => ?_⟩ |
| exact (FiniteMeasure.tendsto_iff_forall_integral_tendsto.mp ht) (BoundedContinuousFunction.mkOfCompact f) |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
