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67aee1d | 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 | 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
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