File size: 3,457 Bytes
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 | import OAI.Analysis.FiniteFactor.Sampling.SecondMoment
namespace OAI
namespace FiniteFactor
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal
noncomputable section
/-- At each retained starting coordinate, the source weight is exactly the
law of the displacement v times the distance to the endpoint. -/
theorem samplingWeight_displacement (h α z : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α)
(hz : z ∈ Set.Ico (0 : ℝ) 1) (hcut : α ≤ samplingDistance z) :
unitIntervalMeasure.withDensity (fun t => ENNReal.ofReal (samplingWeight false h α z t)) =
(samplingVelocityMeasure h).map (fun v => samplingDistance z * v) := by
let d := samplingDistance z
have hd : 0 < d := hα.trans_le hcut
have hL : 0 < h*d := mul_pos hh hd
have hd1 : d ≤ 1-z := min_le_right _ _
have hL1 : h*d ≤ 1 := by nlinarith [hz.1]
have he : (fun t => ENNReal.ofReal (samplingWeight false h α z t)) =
(Set.Ioo 0 (h*d)).indicator (fun _ => ENNReal.ofReal (1/(h*d))) := by
funext t
simp only [samplingWeight,Bool.false_eq_true,ite_false,hcut,true_and,Set.indicator,Set.mem_Ioo,d]
split_ifs <;> simp
rw [he,withDensity_indicator measurableSet_Ioo,withDensity_const]
unfold unitIntervalMeasure
rw [Measure.restrict_restrict measurableSet_Ioo,
Set.inter_eq_left.mpr (show Set.Ioo 0 (h*d) ⊆ Set.Ico (0 : ℝ) 1 from
fun t ht => ⟨ht.1.le,ht.2.trans_le hL1⟩),restrict_Ioo_eq_restrict_Ioc]
unfold samplingVelocityMeasure ProbabilityTheory.cond
rw [Measure.map_smul _ (by fun_prop)]
have ha := map_interval_affine 0 h d 0 hd
simp only [mul_zero,add_zero] at ha
rw [ha,smul_smul,Real.volume_Ioc,sub_zero]
congr 1
rw [← ENNReal.ofReal_inv_of_pos hh,← ENNReal.ofReal_mul (inv_nonneg.mpr hh.le)]
congr 1
dsimp [d]
field_simp
simp [mul_comm]
/-- Exact substitution t=v*Delta(z), at each retained starting point. -/
theorem samplingWeight_transport_integral (h α z : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α)
(hz : z ∈ Set.Ico (0 : ℝ) 1) (hcut : α ≤ samplingDistance z)
(F : ℝ → ℝ≥0∞) (hF : Measurable F) :
(∫⁻ t, ENNReal.ofReal (samplingWeight false h α z t)*F (z+t) ∂unitIntervalMeasure) =
∫⁻ v, F (samplingTransport v z) ∂samplingVelocityMeasure h := by
have hw : Measurable (fun t => ENNReal.ofReal (samplingWeight false h α z t)) :=
((samplingWeight_measurable false h α).comp (measurable_const.prodMk measurable_id)).ennreal_ofReal
have hFz : Measurable (fun t => F (z+t)) := hF.comp (by fun_prop)
have hi := lintegral_withDensity_eq_lintegral_mul unitIntervalMeasure hw hFz
simp only [Pi.mul_apply] at hi
rw [← hi,
samplingWeight_displacement h α z hh hh1 hα hz hcut,
lintegral_map hFz (by fun_prop)]
apply lintegral_congr
intro v
simp [samplingTransport,mul_comm]
/-- Discarding the cutoff only increases the weighted endpoint law. -/
theorem samplingWeight_transport_le (h α z : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α)
(hz : z ∈ Set.Ico (0 : ℝ) 1) (F : ℝ → ℝ≥0∞) (hF : Measurable F) :
(∫⁻ t, ENNReal.ofReal (samplingWeight false h α z t)*F (z+t) ∂unitIntervalMeasure) ≤
∫⁻ v, F (samplingTransport v z) ∂samplingVelocityMeasure h := by
by_cases hc : α ≤ samplingDistance z
· exact (samplingWeight_transport_integral h α z hh hh1 hα hz hc F hF).le
· simp [samplingWeight,hc]
end
end FiniteFactor
end OAI
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