File size: 5,886 Bytes
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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | 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]
/-- The chosen real representatives recover the actual orbit measures;
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
/-- Actual compact-torus support: representatives are ordered, and each pair
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
|