| import OAI.Analysis.FiniteFactor.OrbitSampling.TriangleLaw |
| import OAI.Analysis.FiniteFactor.OrbitSampling.IdealRectangleLimit |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
|
|
| abbrev sourceSamplingPairHaar : Measure (SourceCoordinate → ℝ × ℝ) := |
| Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure.prod unitIntervalMeasure) |
|
|
| def sourceSamplingTriangleSet (I : Finset SourceCoordinate) : Set (SourceCoordinate → ℝ × ℝ) := |
| {p | ∀ i ∈ I, (p i).1 < (p i).2} |
|
|
| def sourceSamplingLimitMap (I : Finset SourceCoordinate) (p : SourceCoordinate → ℝ × ℝ) : |
| (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) := |
| (fun i => (p i).1,fun i => if i ∈ I then (p i).2 else (p i).1) |
|
|
| def sourceSamplingLambda (I : Finset SourceCoordinate) : |
| Measure ((SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) := |
| (sourceSamplingPairHaar.restrict (sourceSamplingTriangleSet I)).map (sourceSamplingLimitMap I) |
|
|
| theorem sourceSamplingTriangleSet_measurable (I : Finset SourceCoordinate) : |
| MeasurableSet (sourceSamplingTriangleSet I) := by |
| simp only [sourceSamplingTriangleSet,Set.ofPred_forall] |
| apply MeasurableSet.iInter |
| intro i |
| apply MeasurableSet.iInter |
| intro _ |
| exact measurableSet_lt (measurable_fst.comp (measurable_pi_apply i)) |
| (measurable_snd.comp (measurable_pi_apply i)) |
|
|
| theorem sourceSamplingLimitMap_measurable (I : Finset SourceCoordinate) : |
| Measurable (sourceSamplingLimitMap I) := by |
| apply Measurable.prodMk |
| · exact Measurable.of_eval (fun i => measurable_fst.comp (measurable_pi_apply i)) |
| · apply Measurable.of_eval |
| intro i |
| by_cases hi : i ∈ I |
| · simpa only [hi,ite_true,Function.comp_def] using measurable_snd.comp (measurable_pi_apply i) |
| · simpa only [hi,ite_false,Function.comp_def] using measurable_fst.comp (measurable_pi_apply i) |
|
|
| instance sourceSamplingLambda_finite (I : Finset SourceCoordinate) : IsFiniteMeasure (sourceSamplingLambda I) := by |
| unfold sourceSamplingLambda |
| infer_instance |
|
|
| theorem sourceSamplingPairHaar_cube : ∀ᵐ p ∂sourceSamplingPairHaar, |
| ∀ i, (p i).1 ∈ Set.Ico (0 : ℝ) 1 ∧ (p i).2 ∈ Set.Ico (0 : ℝ) 1 := by |
| apply ae_all_iff.mpr |
| intro i |
| have hp : ∀ᵐ z ∂unitIntervalMeasure.prod unitIntervalMeasure, |
| z.1 ∈ Set.Ico (0 : ℝ) 1 ∧ z.2 ∈ Set.Ico (0 : ℝ) 1 := by |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_Ico.prod measurableSet_Ico)).mpr |
| filter_upwards [ae_restrict_mem measurableSet_Ico] with x hx |
| filter_upwards [ae_restrict_mem measurableSet_Ico] with y hy |
| exact ⟨hx,hy⟩ |
| have hh : ∀ᵐ z ∂sourceSamplingPairHaar.map (fun p => p i), |
| z.1 ∈ Set.Ico (0 : ℝ) 1 ∧ z.2 ∈ Set.Ico (0 : ℝ) 1 := by |
| rw [Measure.infinitePi_map_eval] |
| exact hp |
| exact ae_of_ae_map (measurable_pi_apply i).aemeasurable hh |
|
|
| /-- The actual limiting measure is carried by ordered pairs. -/ |
| theorem sourceSamplingLambda_ordered (I : Finset SourceCoordinate) : |
| ∀ᵐ p ∂sourceSamplingLambda I, ∀ i, p.1 i ≤ p.2 i := by |
| have horder : MeasurableSet {p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) | |
| ∀ i, p.1 i ≤ p.2 i} := by |
| simp only [Set.ofPred_forall] |
| exact MeasurableSet.iInter (fun i => measurableSet_le |
| ((measurable_pi_apply i).comp measurable_fst) ((measurable_pi_apply i).comp measurable_snd)) |
| apply (ae_map_iff (sourceSamplingLimitMap_measurable I).aemeasurable horder).mpr |
| filter_upwards [ae_restrict_mem (sourceSamplingTriangleSet_measurable I)] with p hp |
| intro i |
| by_cases hi : i ∈ I |
| · simpa only [sourceSamplingLimitMap,ite_eq_left hi] using (hp i hi).le |
| · simp [sourceSamplingLimitMap,hi] |
|
|
| /-- The finite rectangle formula for the source limit, written using actual |
| Lebesgue triangular and diagonal masses. -/ |
| theorem sourceSamplingLimitRectangleMass_formula (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest) : |
| sourceSamplingLimitRectangleMass I T = ∏ i ∈ I ∪ T.1, |
| if i ∈ I then ((unitIntervalMeasure.prod unitIntervalMeasure) |
| {p : ℝ × ℝ | p.1 ∈ sourceSamplingRationalX T i ∧ |
| p.2 ∈ sourceSamplingRationalY T i ∧ p.1 < p.2}).toReal |
| else (unitIntervalMeasure (sourceSamplingRationalX T i ∩ sourceSamplingRationalY T i)).toReal := by |
| apply Finset.prod_congr rfl |
| intro i _ |
| by_cases hi : i ∈ I |
| · simp only [ite_eq_left hi] |
| exact samplingTestedFirstDensity_triangular_mass _ _ _ _ measurableSet_Icc measurableSet_Icc |
| · simp only [ite_eq_right hi] |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
