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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Analysis.FiniteFactor.OrbitSampling.LimitMeasure
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
def sourceSamplingLambdaTest (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest)
(i : SourceCoordinate) (p : ℝ × ℝ) : Prop :=
if i ∈ I then p.1 ∈ sourceSamplingRationalX T i ∧ p.2 ∈ sourceSamplingRationalY T i ∧ p.1 < p.2
else p.1 ∈ sourceSamplingRationalX T i ∧ p.1 ∈ sourceSamplingRationalY T i
theorem sourceSamplingLambdaTest_measurable (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest)
(i : SourceCoordinate) : MeasurableSet {p | sourceSamplingLambdaTest I T i p} := by
by_cases hi : i ∈ I
· simp only [sourceSamplingLambdaTest,ite_eq_left hi]
exact (measurableSet_Icc.preimage measurable_fst).inter
((measurableSet_Icc.preimage measurable_snd).inter (measurableSet_lt measurable_fst measurable_snd))
· simp only [sourceSamplingLambdaTest,ite_eq_right hi]
exact (measurableSet_Icc.preimage measurable_fst).inter (measurableSet_Icc.preimage measurable_fst)
/-- The finite coordinate conditions are equivalent to restriction by the
triangle and the full cylinder rectangle on the conull Haar cube. -/
theorem sourceSamplingLambdaTest_iff (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest)
(p : SourceCoordinate → ℝ × ℝ)
(hp : ∀ i, (p i).1 ∈ Set.Ico (0 : ℝ) 1 ∧ (p i).2 ∈ Set.Ico (0 : ℝ) 1) :
(∀ i ∈ I ∪ T.1, sourceSamplingLambdaTest I T i (p i)) ↔
sourceSamplingLimitMap I p ∈ sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T) ∧
p ∈ sourceSamplingTriangleSet I := by
constructor
· intro h
constructor
· intro i
by_cases hi : i ∈ I
· have hh := h i (Finset.mem_union_left _ hi)
simp only [sourceSamplingLambdaTest,ite_eq_left hi] at hh
exact ⟨hh.1,by simpa only [sourceSamplingLimitMap,ite_eq_left hi] using hh.2.1⟩
· by_cases ht : i ∈ T.1
· have hh := h i (Finset.mem_union_right _ ht)
simpa only [sourceSamplingLambdaTest,sourceSamplingLimitMap,ite_eq_right hi] using hh
· have hx : (p i).1 ∈ Set.Icc (0 : ℝ) 1 := ⟨(hp i).1.1,(hp i).1.2.le⟩
simpa only [sourceSamplingLimitMap,ite_eq_right hi,sourceSamplingRationalX,sourceSamplingRationalY,
sourceSamplingRationalEndpoints,dite_eq_right ht,Rat.cast_zero,Rat.cast_one] using And.intro hx hx
· intro i hi
have hh := h i (Finset.mem_union_left _ hi)
simp only [sourceSamplingLambdaTest,ite_eq_left hi] at hh
exact hh.2.2
· rintro ⟨hrect,htri⟩ i _
have hh := hrect i
by_cases hi : i ∈ I
· simp only [sourceSamplingLambdaTest,ite_eq_left hi]
exact ⟨hh.1,by simpa only [sourceSamplingLimitMap,ite_eq_left hi] using hh.2,htri i hi⟩
· simpa only [sourceSamplingLambdaTest,sourceSamplingLimitMap,ite_eq_right hi] using hh
/-- The constructed infinite-coordinate measure has exactly the rectangle
values previously obtained as sampling limits. -/
theorem sourceSamplingLambda_rectangle (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest) :
((sourceSamplingLambda I)
(sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T))).toReal =
sourceSamplingLimitRectangleMass I T := by
let R := sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T)
let C := sourceSamplingLimitMap I ⁻¹' R ∩ sourceSamplingTriangleSet I
have hR : MeasurableSet R := sourceSamplingRectangle_measurable _ _
(fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc)
have hmap := sourceSamplingLimitMap_measurable I
have hC : MeasurableSet C := (hmap hR).inter (sourceSamplingTriangleSet_measurable I)
have he : C.indicator (fun _ => (1 : ℝ)) =ᵐ[sourceSamplingPairHaar]
(fun p => ∏ i ∈ I ∪ T.1, if sourceSamplingLambdaTest I T i (p i) then (1 : ℝ) else 0) := by
filter_upwards [sourceSamplingPairHaar_cube] with p hp
simp only [Finset.prod_ite_zero,Finset.prod_const_one,
sourceSamplingLambdaTest_iff I T p hp,Set.indicator_apply]
rfl
rw [sourceSamplingLambda,Measure.map_apply hmap hR,Measure.restrict_apply (hmap hR)]
change (sourceSamplingPairHaar C).toReal = _
calc
_ = ∫ p, C.indicator (fun _ => (1 : ℝ)) p ∂sourceSamplingPairHaar := by
rw [integral_indicator hC]; simp [measureReal_def]
_ = ∫ p, ∏ i ∈ I ∪ T.1, if sourceSamplingLambdaTest I T i (p i) then (1 : ℝ) else 0
∂sourceSamplingPairHaar := integral_congr_ae he
_ = ∏ i ∈ I ∪ T.1, ∫ z, (if sourceSamplingLambdaTest I T i z then (1 : ℝ) else 0)
∂unitIntervalMeasure.prod unitIntervalMeasure :=
integral_infinitePi_finset_prod _ _ _ (fun i =>
Measurable.ite (sourceSamplingLambdaTest_measurable I T i) measurable_const measurable_const)
_ = _ := by
rw [sourceSamplingLimitRectangleMass_formula]
apply Finset.prod_congr rfl
intro i _
have hA := sourceSamplingLambdaTest_measurable I T i
have hh : (∫ z, (if sourceSamplingLambdaTest I T i z then (1 : ℝ) else 0)
∂unitIntervalMeasure.prod unitIntervalMeasure) =
((unitIntervalMeasure.prod unitIntervalMeasure) {z | sourceSamplingLambdaTest I T i z}).toReal := by
change (∫ z, {z | sourceSamplingLambdaTest I T i z}.indicator (fun _ => (1 : ℝ)) z
∂unitIntervalMeasure.prod unitIntervalMeasure) = _
rw [integral_indicator hA]; simp [measureReal_def]
rw [hh]
by_cases hi : i ∈ I
· simp only [sourceSamplingLambdaTest,ite_eq_left hi]
· simp only [sourceSamplingLambdaTest,ite_eq_right hi]
have heq : {z : ℝ × ℝ | z.1 ∈ sourceSamplingRationalX T i ∧ z.1 ∈ sourceSamplingRationalY T i} =
(sourceSamplingRationalX T i ∩ sourceSamplingRationalY T i) ×ˢ Set.univ := by ext z; simp
rw [heq,Measure.prod_prod]
simp
/-- Actual untrimmed measures converge to the actual source limiting measure
on every rational cylinder rectangle, on a single conull event. -/
theorem sourceSamplingLambda_rectangle_limit_event :
∀ᵐ θ ∂sourceAngleLaw, ∀ (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest),
Tendsto (fun stage => ((sourceSamplingUntrimmed I stage θ)
(sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T))).toReal)
atTop (𝓝 (((sourceSamplingLambda I)
(sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T))).toReal)) := by
filter_upwards [sourceSampling_rectangle_limit_event] with θ hθ
intro I T
rw [sourceSamplingLambda_rectangle]
exact hθ I T
end
end FiniteFactor
end OAI
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