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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 | import OAI.Analysis.FiniteFactor.Sampling.SelfMoment
import OAI.Analysis.FiniteFactor.Sampling.TestedSecondMoment
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
noncomputable section
/-- A bounded measurable real function has an integrable square on a finite measure. -/
theorem sampling_square_integrable {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
[IsFiniteMeasure μ] (f : Ω → ℝ) (C : ℝ) (hm : Measurable f)
(hb : ∀ x, 0 ≤ f x ∧ f x ≤ C) : Integrable (fun x => f x ^ 2) μ := by
apply (integrable_const (C^2)).mono' (hm.pow_const 2).aestronglyMeasurable
exact Eventually.of_forall (fun x => by
rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg (f x))]
nlinarith [(hb x).1,(hb x).2])
/-- Squared weights are integrable jointly, permitting the source's Fubini step. -/
theorem samplingWeight_square_integrable (b : Bool) (h α : ℝ) (hh : 0 < h) (hα : 0 < α) :
Integrable (fun p : ℝ × ℝ => samplingWeight b h α p.1 p.2 ^ 2)
(unitIntervalMeasure.prod unitIntervalMeasure) :=
sampling_square_integrable _ _ (max 1 (1/(h*α))) (samplingWeight_measurable b h α)
(fun p => samplingWeight_bounds b h α p.1 p.2 hh hα)
/-- Haar translation preserves the untested self-pair moment in the second
orientation, with the source's order of both integrations. -/
theorem samplingSecondWeight_square_integral (h α : ℝ) (hh : 0 < h) (hh1 : h < 1)
(hα : 0 < α) (hα1 : α ≤ 1/2) :
(∫ s, ∫ t, samplingSecondWeight h α s t ^ 2
∂unitIntervalMeasure ∂unitIntervalMeasure) = (2/h)*Real.log (1/(2*α)) := by
have hi : Integrable (fun p : ℝ × ℝ => samplingSecondWeight h α p.1 p.2 ^ 2)
(unitIntervalMeasure.prod unitIntervalMeasure) :=
sampling_square_integrable _ _ (max 1 (1/(h*α))) (samplingSecondWeight_measurable h α)
(fun p => samplingWeight_bounds false h α (Int.fract (p.1-p.2)) p.2 hh hα)
rw [integral_integral_swap hi]
calc
_ = ∫ t, ∫ s, samplingWeight false h α s t ^ 2 ∂unitIntervalMeasure ∂unitIntervalMeasure := by
apply integral_congr_ae
filter_upwards [] with t
have hm : AEStronglyMeasurable (fun s => samplingWeight false h α s t ^ 2) unitIntervalMeasure :=
(((samplingWeight_measurable false h α).comp (measurable_id.prodMk measurable_const)).pow_const 2).aestronglyMeasurable
have hp := unitInterval_rotate_preserving (-t)
have he := integral_map hp.measurable.aemeasurable (show AEStronglyMeasurable
(fun s => samplingWeight false h α s t ^ 2) (unitIntervalMeasure.map (fun s => Int.fract (s+ -t))) from
hp.map_eq.symm ▸ hm)
rw [hp.map_eq] at he
simpa [samplingSecondWeight,sub_eq_add_neg] using he.symm
_ = _ := by
rw [← integral_integral_swap (samplingWeight_square_integrable false h α hh hα),
samplingWeight_square_integral h α hh hh1 hα hα1]
/-- Tests decrease the diagonal self-pair moment in the first orientation. -/
theorem samplingTestedFirstWeight_square_integral (h α : ℝ) (E G : Set ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hα1 : α ≤ 1/2)
(hE : MeasurableSet E) (hG : MeasurableSet G) :
(∫ s, ∫ t, samplingTestedFirstWeight false h α E G s t ^ 2
∂unitIntervalMeasure ∂unitIntervalMeasure) ≤ (2/h)*Real.log (1/(2*α)) := by
have hb (s t : ℝ) := samplingTestedFirstWeight_bounds false h α E G s t hh hα
have hm := samplingTestedFirstWeight_measurable false h α E G hE hG
have hbig := samplingWeight_square_integrable false h α hh hα
have hsmall : Integrable (fun p : ℝ × ℝ => samplingTestedFirstWeight false h α E G p.1 p.2 ^ 2)
(unitIntervalMeasure.prod unitIntervalMeasure) :=
sampling_square_integrable _ _ (max 1 (1/(h*α))) hm
(fun p => ⟨(hb p.1 p.2).1,(hb p.1 p.2).2.trans (samplingWeight_bounds false h α p.1 p.2 hh hα).2⟩)
rw [← samplingWeight_square_integral h α hh hh1 hα hα1,
← integral_prod _ hsmall,← integral_prod _ hbig]
apply integral_mono_ae hsmall hbig
exact Eventually.of_forall (fun p => by
nlinarith [(hb p.1 p.2).1,(hb p.1 p.2).2,
(samplingWeight_bounds false h α p.1 p.2 hh hα).1])
/-- Tests decrease the diagonal self-pair moment in the second orientation. -/
theorem samplingTestedSecondWeight_square_integral (h α : ℝ) (E G : Set ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hα1 : α ≤ 1/2)
(hE : MeasurableSet E) (hG : MeasurableSet G) :
(∫ s, ∫ t, samplingTestedSecondWeight h α E G s t ^ 2
∂unitIntervalMeasure ∂unitIntervalMeasure) ≤ (2/h)*Real.log (1/(2*α)) := by
have hb (s t : ℝ) := samplingTestedSecondWeight_bounds h α E G s t hh hα
have hm := samplingTestedSecondWeight_measurable h α E G hE hG
have hbig : Integrable (fun p : ℝ × ℝ => samplingSecondWeight h α p.1 p.2 ^ 2)
(unitIntervalMeasure.prod unitIntervalMeasure) :=
sampling_square_integrable _ _ (max 1 (1/(h*α))) (samplingSecondWeight_measurable h α)
(fun p => samplingWeight_bounds false h α (Int.fract (p.1-p.2)) p.2 hh hα)
have hsmall : Integrable (fun p : ℝ × ℝ => samplingTestedSecondWeight h α E G p.1 p.2 ^ 2)
(unitIntervalMeasure.prod unitIntervalMeasure) :=
sampling_square_integrable _ _ (max 1 (1/(h*α))) hm
(fun p => ⟨(hb p.1 p.2).1,(hb p.1 p.2).2.trans
(samplingWeight_bounds false h α (Int.fract (p.1-p.2)) p.2 hh hα).2⟩)
rw [← samplingSecondWeight_square_integral h α hh hh1 hα hα1,
← integral_prod _ hsmall,← integral_prod _ hbig]
apply integral_mono_ae hsmall hbig
exact Eventually.of_forall (fun p => by
nlinarith [(hb p.1 p.2).1,(hb p.1 p.2).2,
(samplingWeight_bounds false h α (Int.fract (p.1-p.2)) p.2 hh hα).1])
end
end FiniteFactor
end OAI
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