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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 | import OAI.Analysis.FiniteFactor.Sampling.AveragedTransport
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
namespace OAI
namespace FiniteFactor
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal
noncomputable section
def samplingSecondWeight (h α s t : ℝ) : ℝ :=
samplingWeight false h α (Int.fract (s-t)) t
def samplingSecondDensity (h α s : ℝ) : ℝ :=
∫ t, samplingSecondWeight h α s t ∂unitIntervalMeasure
theorem samplingSecondWeight_measurable (h α : ℝ) :
Measurable (fun p : ℝ × ℝ => samplingSecondWeight h α p.1 p.2) :=
(samplingWeight_measurable false h α).comp
((sourceFract_measurable.comp (measurable_fst.sub measurable_snd)).prodMk measurable_snd)
theorem samplingSecondWeight_integrable (h α s : ℝ) (hh : 0 < h) (hα : 0 < α) :
Integrable (samplingSecondWeight h α s) unitIntervalMeasure := by
have hm : Measurable (samplingSecondWeight h α s) :=
(samplingSecondWeight_measurable h α).comp (measurable_const.prodMk measurable_id)
exact (integrable_const (max 1 (1/(h*α)))).mono' hm.aestronglyMeasurable
(Eventually.of_forall (fun t => by
rw [Real.norm_eq_abs]
change |samplingWeight false h α (Int.fract (s-t)) t| ≤ _
rw [abs_of_nonneg (samplingWeight_bounds false h α (Int.fract (s-t)) t hh hα).1]
exact (samplingWeight_bounds false h α _ t hh hα).2))
theorem samplingSecondDensity_lintegral (h α s : ℝ) (hh : 0 < h) (hα : 0 < α) :
ENNReal.ofReal (samplingSecondDensity h α s) =
∫⁻ t, ENNReal.ofReal (samplingSecondWeight h α s t) ∂unitIntervalMeasure :=
ofReal_integral_eq_lintegral_ofReal (samplingSecondWeight_integrable h α s hh hα)
(Eventually.of_forall (fun t => (samplingWeight_bounds false h α _ t hh hα).1))
theorem samplingSecondDensity_measurable (h α : ℝ) : Measurable (samplingSecondDensity h α) :=
(samplingSecondWeight_measurable h α).stronglyMeasurable.integral_prod_right.measurable
theorem samplingSecondDensity_nonneg (h α s : ℝ) (hh : 0 < h) (hα : 0 < α) :
0 ≤ samplingSecondDensity h α s :=
integral_nonneg (fun t => (samplingWeight_bounds false h α _ t hh hα).1)
theorem sampling_fract_inverse (z t : ℝ) (hz : z ∈ Set.Ico (0 : ℝ) 1) :
Int.fract (Int.fract (z+t)-t) = z := by
have he : Int.fract (z+t)-t = z-(Int.floor (z+t) : ℝ) := by rw [Int.fract]; ring
rw [he,Int.fract_sub_intCast,Int.fract_eq_self.mpr hz]
/-- Haar substitution identifies the actual second-oriented integral with
the weighted endpoint law; no marginal-density hypothesis is assumed. -/
theorem samplingSecondDensity_test (h α : ℝ) (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α)
(F : ℝ → ℝ≥0∞) (hF : Measurable F) :
(∫⁻ s, ENNReal.ofReal (samplingSecondDensity h α s)*F s ∂unitIntervalMeasure) =
∫⁻ z, ∫⁻ t, ENNReal.ofReal (samplingWeight false h α z t)*F (z+t)
∂unitIntervalMeasure ∂unitIntervalMeasure := by
have hwm := samplingSecondWeight_measurable h α
calc
_ = ∫⁻ s, ∫⁻ t, ENNReal.ofReal (samplingSecondWeight h α s t)*F s
∂unitIntervalMeasure ∂unitIntervalMeasure := by
apply lintegral_congr
intro s
have hmd : Measurable (fun t => ENNReal.ofReal (samplingSecondWeight h α s t)) :=
(hwm.comp (measurable_const.prodMk measurable_id)).ennreal_ofReal
rw [samplingSecondDensity_lintegral h α s hh hα,lintegral_mul_const _ hmd]
_ = ∫⁻ t, ∫⁻ s, ENNReal.ofReal (samplingSecondWeight h α s t)*F s
∂unitIntervalMeasure ∂unitIntervalMeasure :=
lintegral_lintegral_swap (hwm.ennreal_ofReal.mul (hF.comp measurable_fst)).aemeasurable
_ = ∫⁻ t, ∫⁻ z, ENNReal.ofReal (samplingWeight false h α z t)*F (Int.fract (z+t))
∂unitIntervalMeasure ∂unitIntervalMeasure := by
apply lintegral_congr
intro t
have hm : Measurable (fun s => ENNReal.ofReal (samplingSecondWeight h α s t)*F s) :=
((hwm.comp (measurable_id.prodMk measurable_const)).ennreal_ofReal).mul hF
rw [← (unitInterval_rotate_preserving t).lintegral_comp hm]
apply lintegral_congr_ae
filter_upwards [show ∀ᵐ z ∂unitIntervalMeasure, z ∈ Set.Ico (0 : ℝ) 1 from
ae_restrict_mem measurableSet_Ico] with z hz
simp only [samplingSecondWeight,sampling_fract_inverse z t hz]
_ = ∫⁻ z, ∫⁻ t, ENNReal.ofReal (samplingWeight false h α z t)*F (Int.fract (z+t))
∂unitIntervalMeasure ∂unitIntervalMeasure := by
apply lintegral_lintegral_swap
exact (((samplingWeight_measurable false h α).comp measurable_swap).ennreal_ofReal.mul
(hF.comp (sourceFract_measurable.comp (measurable_snd.add measurable_fst)))).aemeasurable
_ = _ := by
apply lintegral_congr_ae
filter_upwards [show ∀ᵐ z ∂unitIntervalMeasure, z ∈ Set.Ico (0 : ℝ) 1 from
ae_restrict_mem measurableSet_Ico] with z hz
apply lintegral_congr
intro t
by_cases hw : samplingWeight false h α z t = 0
· simp [hw]
· rw [(samplingWeight_orders false h α z t hh hh1 hα hz hw).1]
/-- The actual cutoff second marginal is dominated almost everywhere by
q_h, completing the missing connection to the quadratic ideal moment. -/
theorem samplingSecondDensity_le_ideal (h α : ℝ) (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) :
∀ᵐ s ∂unitIntervalMeasure, samplingSecondDensity h α s ≤ samplingIdealSecond h s := by
have he : ∀ᵐ s ∂unitIntervalMeasure,
ENNReal.ofReal (samplingSecondDensity h α s) ≤ ENNReal.ofReal (samplingIdealSecond h s) := by
apply ae_le_of_forall_setLIntegral_le_of_sigmaFinite
(samplingSecondDensity_measurable h α).ennreal_ofReal
intro S hS _
have hb := samplingWeight_endpoint_domination h α hh hh1 hα (S.indicator (fun _ => 1))
(measurable_const.indicator hS)
rw [← samplingSecondDensity_test h α hh hh1 hα _ (measurable_const.indicator hS)] at hb
have hind (g : ℝ → ℝ≥0∞) : (fun s => g s*S.indicator (fun _ => 1) s) = S.indicator g := by
funext s
by_cases hs : s ∈ S <;> simp [Set.indicator,hs]
rw [hind,hind,lintegral_indicator hS,lintegral_indicator hS] at hb
exact hb
filter_upwards [he] with s hs
exact (ENNReal.ofReal_le_ofReal_iff (samplingIdealSecond_bounds h s hh hh1).1).mp hs
end
end FiniteFactor
end OAI
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