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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 | import OAI.Analysis.FiniteFactor.Sampling.Weights
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
noncomputable section
def samplingTestedFirstWeight (b : Bool) (h α : ℝ) (E G : Set ℝ) (s t : ℝ) : ℝ := by
classical
exact if s ∈ E ∧ Int.fract (s+t) ∈ G then samplingWeight b h α s t else 0
def samplingTestedFirstDensity (b : Bool) (h α : ℝ) (E G : Set ℝ) (s : ℝ) : ℝ :=
∫ t, samplingTestedFirstWeight b h α E G s t ∂unitIntervalMeasure
theorem samplingTestedFirstWeight_measurable (b : Bool) (h α : ℝ) (E G : Set ℝ)
(hE : MeasurableSet E) (hG : MeasurableSet G) :
Measurable (fun p : ℝ × ℝ => samplingTestedFirstWeight b h α E G p.1 p.2) := by
unfold samplingTestedFirstWeight
exact Measurable.ite ((hE.preimage measurable_fst).inter
(hG.preimage (sourceFract_measurable.comp (measurable_fst.add measurable_snd))))
(samplingWeight_measurable b h α) measurable_const
theorem samplingTestedFirstWeight_bounds (b : Bool) (h α : ℝ) (E G : Set ℝ) (s t : ℝ)
(hh : 0 < h) (hα : 0 < α) :
0 ≤ samplingTestedFirstWeight b h α E G s t ∧
samplingTestedFirstWeight b h α E G s t ≤ samplingWeight b h α s t := by
have hn := (samplingWeight_bounds b h α s t hh hα).1
unfold samplingTestedFirstWeight
split_ifs <;> simp [hn]
theorem samplingWeight_integrable (b : Bool) (h α s : ℝ) (hh : 0 < h) (hα : 0 < α) :
Integrable (samplingWeight b h α s) unitIntervalMeasure := by
apply (integrable_const (max 1 (1/(h*α)))).mono'
((samplingWeight_measurable b h α).comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable
exact Eventually.of_forall (fun t => by
change ‖samplingWeight b h α s t‖ ≤ _
rw [Real.norm_eq_abs,abs_of_nonneg (samplingWeight_bounds b h α s t hh hα).1]
exact (samplingWeight_bounds b h α s t hh hα).2)
theorem samplingTestedFirstDensity_measurable (b : Bool) (h α : ℝ) (E G : Set ℝ)
(hE : MeasurableSet E) (hG : MeasurableSet G) :
Measurable (samplingTestedFirstDensity b h α E G) :=
(samplingTestedFirstWeight_measurable b h α E G hE hG).stronglyMeasurable.integral_prod_right.measurable
/-- Tests decrease the exact first marginal, which is at most one. -/
theorem samplingTestedFirstDensity_bounds (b : Bool) (h α : ℝ) (E G : Set ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hE : MeasurableSet E) (hG : MeasurableSet G)
(s : ℝ) (hs : s ∈ Set.Ico (0 : ℝ) 1) :
0 ≤ samplingTestedFirstDensity b h α E G s ∧
samplingTestedFirstDensity b h α E G s ≤ 1 := by
have hm : Measurable (samplingTestedFirstWeight b h α E G s) := (samplingTestedFirstWeight_measurable b h α E G hE hG).comp
(measurable_const.prodMk measurable_id)
have hi := (samplingWeight_integrable b h α s hh hα).mono' hm.aestronglyMeasurable
(Eventually.of_forall (fun t => by
rw [Real.norm_eq_abs,abs_of_nonneg (samplingTestedFirstWeight_bounds b h α E G s t hh hα).1]
exact (samplingTestedFirstWeight_bounds b h α E G s t hh hα).2))
refine ⟨integral_nonneg (fun t => (samplingTestedFirstWeight_bounds b h α E G s t hh hα).1), ?_⟩
calc
_ ≤ ∫ t, samplingWeight b h α s t ∂unitIntervalMeasure :=
integral_mono_ae hi (samplingWeight_integrable b h α s hh hα)
(Eventually.of_forall (fun t => (samplingTestedFirstWeight_bounds b h α E G s t hh hα).2))
_ ≤ 1 := by
rw [samplingWeight_first_integral b h α s hh hh1 hα hs]
cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true]
· split_ifs <;> norm_num
· linarith [hs.1]
/-- The first tested marginal has quadratic moment at most one in both
triangular and diagonal coordinates, uniformly in the measurable tests. -/
theorem samplingTestedFirstDensity_square (b : Bool) (h α : ℝ) (E G : Set ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hE : MeasurableSet E) (hG : MeasurableSet G) :
(∫ s, samplingTestedFirstDensity b h α E G s ^ 2 ∂unitIntervalMeasure) ≤ 1 := by
have hb : ∀ᵐ s ∂unitIntervalMeasure, 0 ≤ samplingTestedFirstDensity b h α E G s ∧
samplingTestedFirstDensity b h α E G s ≤ 1 :=
(ae_restrict_mem measurableSet_Ico).mono (fun s hs =>
samplingTestedFirstDensity_bounds b h α E G hh hh1 hα hE hG s hs)
have hb2 := hb.mono (fun s hs => show samplingTestedFirstDensity b h α E G s ^ 2 ≤ 1 by nlinarith)
have hi : Integrable (fun s => samplingTestedFirstDensity b h α E G s ^ 2) unitIntervalMeasure :=
(integrable_const (1 : ℝ)).mono'
((samplingTestedFirstDensity_measurable b h α E G hE hG).pow_const 2).aestronglyMeasurable
(hb2.mono (fun s hs => by
rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg (samplingTestedFirstDensity b h α E G s))]
exact hs))
simpa using integral_mono_ae hi (integrable_const (1 : ℝ)) hb2
end
end FiniteFactor
end OAI
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