| import OAI.Analysis.FiniteFactor.OrbitSampling.OrderedRestriction | |
| namespace OAI | |
| namespace FiniteFactor | |
| open MeasureTheory Filter Finset | |
| open scoped Classical ENNReal Topology | |
| noncomputable section | |
| def sourceSamplingPositiveOrbits (s : ℕ) (θ : SourceCoordinate → ℝ) : Set SourceSamplingPair := | |
| {p | ∃ n ∈ sourceSamplingPowers s, 0 < n ∧ p.2 = coordinateRotate θ n p.1} | |
| theorem sourceSamplingPositiveOrbits_measurable (s : ℕ) (θ : SourceCoordinate → ℝ) : | |
| MeasurableSet (sourceSamplingPositiveOrbits s θ) := by | |
| simp only [sourceSamplingPositiveOrbits,Set.ofPred_exists] | |
| apply MeasurableSet.iUnion | |
| intro n | |
| by_cases hn : n ∈ sourceSamplingPowers s | |
| · simp only [hn,true_and] | |
| by_cases hp : 0 < n | |
| · simp only [hp,true_and] | |
| exact measurableSet_eq_fun measurable_snd ((coordinateRotate_preserving θ n).measurable.comp measurable_fst) | |
| · simp [hp] | |
| · simp [hn] | |
| theorem sourceSamplingSummand_positive_orbit (I : Finset SourceCoordinate) (s : ℕ) | |
| (θ : SourceCoordinate → ℝ) (n : ℤ) (hn : n ∈ sourceSamplingPowers s) : | |
| ∀ᵐ p ∂sourceSamplingSummand I s θ n, p ∈ sourceSamplingPositiveOrbits s θ := by | |
| apply (ae_map_iff (measurable_id.prodMk (coordinateRotate_preserving θ n).measurable).aemeasurable | |
| (sourceSamplingPositiveOrbits_measurable s θ)).mpr | |
| exact Eventually.of_forall (fun x => ⟨n,hn,(sourceSamplingPowers_positive s n hn).1,rfl⟩) | |
| /-- Trimming preserves the constructed finite set of strictly positive | |
| orbit powers while imposing coordinatewise order everywhere. -/ | |
| theorem sourceSamplingTrimmed_orbits (I : Finset SourceCoordinate) (s : ℕ) | |
| (θ : SourceCoordinate → ℝ) : ∀ᵐ p ∂sourceSamplingTrimmed I s θ, | |
| (∀ i, p.1 i ≤ p.2 i) ∧ ∃ n ∈ sourceSamplingPowers s, 0 < n ∧ p.2 = coordinateRotate θ n p.1 := by | |
| have hu : ∀ᵐ p ∂sourceSamplingUntrimmed I s θ, p ∈ sourceSamplingPositiveOrbits s θ := by | |
| apply Measure.ae_smul_measure | |
| rw [ae_iff,Measure.finsetSum_apply] | |
| exact Finset.sum_eq_zero (fun n hn => ae_iff.mp (sourceSamplingSummand_positive_orbit I s θ n hn)) | |
| filter_upwards [sourceSamplingTrimmed_ordered I s θ,ae_restrict_of_ae hu] with p ho hp | |
| exact ⟨ho,hp⟩ | |
| end | |
| end FiniteFactor | |
| end OAI | |