| import OAI.Analysis.FiniteFactor.CrossedProduct.TorusModel |
| import OAI.Analysis.FiniteFactor.OrbitSampling.OrbitSupport |
| import OAI.Analysis.FiniteFactor.OrbitSampling.TrimmedWeak |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped ENNReal Classical |
| noncomputable section |
|
|
| def sourceSamplingPairRepresentative (z : SourceSamplingTorus) : SourceSamplingPair := |
| (sourceTorusRepresentative z.1,sourceTorusRepresentative z.2) |
|
|
| theorem sourceSamplingPairRepresentative_measurable : Measurable sourceSamplingPairRepresentative := |
| (sourceTorusRepresentative_preserving.measurable.comp measurable_fst).prodMk |
| (sourceTorusRepresentative_preserving.measurable.comp measurable_snd) |
|
|
| def sourceTorusRotate (θ : SourceCoordinate → ℝ) (n : ℤ) (z : SourceTorus) : SourceTorus := |
| fun i => z i + n • (θ i : UnitAddCircle) |
|
|
| theorem sourceTorusRotate_continuous (θ : SourceCoordinate → ℝ) (n : ℤ) : Continuous (sourceTorusRotate θ n) := |
| continuous_pi (fun i => (continuous_apply i).add continuous_const) |
|
|
| theorem sourceTorusQuotient_rotate (θ x : SourceCoordinate → ℝ) (n : ℤ) : |
| sourceTorusQuotient (coordinateRotate θ n x) = sourceTorusRotate θ n (sourceTorusQuotient x) := by |
| funext i |
| simp only [sourceTorusQuotient,coordinateRotate,sourceTorusRotate,AddCircle.coe_fract, |
| AddCircle.coe_add,← zsmul_eq_mul,AddCircle.coe_zsmul] |
|
|
| / |
| closed-cube endpoints are not silently identified with distinct real points. -/ |
| theorem sourceSamplingTrimmed_representatives (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) : ∀ᵐ p ∂sourceSamplingTrimmed I s θ, |
| sourceSamplingPairRepresentative (sourceSamplingTorusMap p) = p := by |
| have hmeas : MeasurableSet {p : SourceSamplingPair | |
| sourceSamplingPairRepresentative (sourceSamplingTorusMap p) = p} := |
| measurableSet_eq_fun (sourceSamplingPairRepresentative_measurable.comp sourceSamplingTorusMap_continuous.measurable) |
| measurable_id |
| have hu : ∀ᵐ p ∂sourceSamplingUntrimmed I s θ, |
| sourceSamplingPairRepresentative (sourceSamplingTorusMap p) = p := by |
| apply Measure.ae_smul_measure |
| rw [ae_iff,Measure.finsetSum_apply] |
| apply Finset.sum_eq_zero |
| intro n hn |
| apply ae_iff.mp |
| apply (ae_map_iff (measurable_id.prodMk (coordinateRotate_preserving θ n).measurable).aemeasurable hmeas).mpr |
| have hh : ∀ᵐ x ∂sourceSamplingWeightedMeasure I s θ n, ∀ i, x i ∈ Set.Ico (0 : ℝ) 1 := |
| ae_iff.mpr (withDensity_absolutelyContinuous _ _ (ae_iff.mp sourceSamplingHaar_cube)) |
| filter_upwards [hh] with x hx |
| apply Prod.ext |
| · exact sourceTorusRepresentative_quotient x hx |
| · exact sourceTorusRepresentative_quotient _ (fun i => ⟨Int.fract_nonneg _,Int.fract_lt_one _⟩) |
| exact ae_restrict_of_ae hu |
|
|
| def sourceSamplingTorusOrderedOrbits (s : ℕ) (θ : SourceCoordinate → ℝ) : Set SourceSamplingTorus := |
| {p | (∀ i, sourceTorusRepresentative p.1 i ≤ sourceTorusRepresentative p.2 i) ∧ |
| ∃ n ∈ sourceSamplingPowers s, 0 < n ∧ p.2 = sourceTorusRotate θ n p.1} |
|
|
| theorem sourceSamplingTorusOrderedOrbits_measurable (s : ℕ) (θ : SourceCoordinate → ℝ) : |
| MeasurableSet (sourceSamplingTorusOrderedOrbits s θ) := by |
| change MeasurableSet ({p : SourceSamplingTorus | ∀ i, sourceTorusRepresentative p.1 i ≤ sourceTorusRepresentative p.2 i} ∩ |
| {p : SourceSamplingTorus | ∃ n ∈ sourceSamplingPowers s, 0 < n ∧ p.2 = sourceTorusRotate θ n p.1}) |
| apply MeasurableSet.inter |
| · simp only [Set.ofPred_forall] |
| exact MeasurableSet.iInter (fun i => measurableSet_le |
| ((measurable_pi_apply i).comp (sourceTorusRepresentative_preserving.measurable.comp measurable_fst)) |
| ((measurable_pi_apply i).comp (sourceTorusRepresentative_preserving.measurable.comp measurable_snd))) |
| · simp only [Set.ofPred_exists] |
| apply MeasurableSet.iUnion |
| intro n |
| by_cases hn : n ∈ sourceSamplingPowers s ∧ 0 < n |
| · simp only [hn.1,hn.2,true_and] |
| exact measurableSet_eq_fun measurable_snd ((sourceTorusRotate_continuous θ n).measurable.comp measurable_fst) |
| · have he : {p : SourceSamplingTorus | n ∈ sourceSamplingPowers s ∧ 0 < n ∧ p.2 = sourceTorusRotate θ n p.1} = ∅ := by |
| ext p; simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]; tauto |
| rw [he] |
| exact MeasurableSet.empty |
|
|
| / |
| lies on one of the constructed finitely many strictly positive rotation graphs. -/ |
| theorem sourceSamplingTrimmed_torus_orbits (I : Finset SourceCoordinate) (s : ℕ) |
| (θ : SourceCoordinate → ℝ) : |
| ∀ᵐ p ∂(sourceSamplingTrimmedTorus I s θ : Measure SourceSamplingTorus), |
| (∀ i, sourceTorusRepresentative p.1 i ≤ sourceTorusRepresentative p.2 i) ∧ |
| ∃ n ∈ sourceSamplingPowers s, 0 < n ∧ p.2 = sourceTorusRotate θ n p.1 := by |
| apply (ae_map_iff sourceSamplingTorusMap_continuous.measurable.aemeasurable |
| (sourceSamplingTorusOrderedOrbits_measurable s θ)).mpr |
| filter_upwards [sourceSamplingTrimmed_orbits I s θ,sourceSamplingTrimmed_representatives I s θ] with p hp he |
| refine ⟨?_,?_⟩ |
| · have h1 := congrArg Prod.fst he |
| have h2 := congrArg Prod.snd he |
| change sourceTorusRepresentative (sourceTorusQuotient p.1) = p.1 at h1 |
| change sourceTorusRepresentative (sourceTorusQuotient p.2) = p.2 at h2 |
| change ∀ i, sourceTorusRepresentative (sourceTorusQuotient p.1) i ≤ sourceTorusRepresentative (sourceTorusQuotient p.2) i |
| rw [h1,h2] |
| exact hp.1 |
| · obtain ⟨n,hn,hpos,hrot⟩ := hp.2 |
| refine ⟨n,hn,hpos,?_⟩ |
| change sourceTorusQuotient p.2 = sourceTorusRotate θ n (sourceTorusQuotient p.1) |
| rw [hrot] |
| exact sourceTorusQuotient_rotate θ p.1 n |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
