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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 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | 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
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