| 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 |
|
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| 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] |
|
|
| / |
| 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] |
|
|
| / |
| 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 |
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| end OAI |
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