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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 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 | import OAI.Analysis.FiniteFactor.OrbitSampling.LambdaRectangles
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
abbrev SourceSamplingPair := (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)
def sourceSamplingCube : Set SourceSamplingPair :=
{p | ∀ i, p.1 i ∈ Set.Icc (0 : ℝ) 1 ∧ p.2 i ∈ Set.Icc (0 : ℝ) 1}
theorem sourceSamplingCube_measurable : MeasurableSet sourceSamplingCube := by
exact sourceSamplingRectangle_measurable _ _ (fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc)
instance sourceSamplingUntrimmed_finite (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) : IsFiniteMeasure (sourceSamplingUntrimmed I s θ) := by
constructor
unfold sourceSamplingUntrimmed
rw [Measure.smul_apply, smul_eq_mul]
exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top (measure_lt_top _ _)
theorem sourceSamplingSummand_cube (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) (n : ℤ) :
∀ᵐ p ∂sourceSamplingSummand I s θ n, p ∈ sourceSamplingCube := by
apply (ae_map_iff (measurable_id.prodMk (coordinateRotate_preserving θ n).measurable).aemeasurable
sourceSamplingCube_measurable).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
intro i
exact ⟨⟨(hx i).1,(hx i).2.le⟩,Int.fract_nonneg _,(Int.fract_lt_one _).le⟩
theorem sourceSamplingUntrimmed_cube (I : Finset SourceCoordinate) (s : ℕ)
(θ : SourceCoordinate → ℝ) :
∀ᵐ p ∂sourceSamplingUntrimmed I s θ, p ∈ sourceSamplingCube := by
apply Measure.ae_smul_measure
rw [ae_iff, Measure.finsetSum_apply]
apply Finset.sum_eq_zero
intro n hn
exact ae_iff.mp (sourceSamplingSummand_cube I s θ n)
theorem sourceSamplingLambda_cube (I : Finset SourceCoordinate) :
∀ᵐ p ∂sourceSamplingLambda I, p ∈ sourceSamplingCube := by
apply (ae_map_iff (sourceSamplingLimitMap_measurable I).aemeasurable
sourceSamplingCube_measurable).mpr
filter_upwards [ae_restrict_of_ae sourceSamplingPairHaar_cube] with p hp
intro i
refine ⟨⟨(hp i).1.1,(hp i).1.2.le⟩,?_⟩
by_cases hi : i ∈ I
· simpa [sourceSamplingLimitMap,hi] using And.intro (hp i).2.1 (hp i).2.2.le
· simpa [sourceSamplingLimitMap,hi] using And.intro (hp i).1.1 (hp i).1.2.le
def sourceSamplingEmptyTest : SourceSamplingRationalTest := ⟨∅,fun _ => ((0,1),(0,1))⟩
theorem sourceSamplingEmptyTest_rectangle :
sourceSamplingRectangle (sourceSamplingRationalX sourceSamplingEmptyTest)
(sourceSamplingRationalY sourceSamplingEmptyTest) = sourceSamplingCube := by
ext p
simp [sourceSamplingRectangle,sourceSamplingRationalX,sourceSamplingRationalY,
sourceSamplingRationalEndpoints,sourceSamplingEmptyTest,sourceSamplingCube]
/-- The empty rational test gives actual total-mass convergence. -/
theorem sourceSampling_mass_limit_event : ∀ᵐ θ ∂sourceAngleLaw,
∀ I : Finset SourceCoordinate,
Tendsto (fun s => ((sourceSamplingUntrimmed I s θ) Set.univ).toReal)
atTop (𝓝 (((sourceSamplingLambda I) Set.univ).toReal)) := by
filter_upwards [sourceSamplingLambda_rectangle_limit_event] with θ hθ
intro I
have h := hθ I sourceSamplingEmptyTest
rw [sourceSamplingEmptyTest_rectangle] at h
have hu (s : ℕ) : sourceSamplingUntrimmed I s θ sourceSamplingCube =
sourceSamplingUntrimmed I s θ Set.univ :=
measure_congr (by filter_upwards [sourceSamplingUntrimmed_cube I s θ] with p hp; exact propext (iff_true_intro hp))
have hl : sourceSamplingLambda I sourceSamplingCube = sourceSamplingLambda I Set.univ :=
measure_congr (by filter_upwards [sourceSamplingLambda_cube I] with p hp; exact propext (iff_true_intro hp))
simpa only [hu,hl] using h
end
end FiniteFactor
end OAI
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