| 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] |
| |
| / |
| 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 |
| |
