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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 | import OAI.Analysis.FiniteFactor.Sampling.SecondLevelAE
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical ENNReal
noncomputable section
theorem sampling_level_pair_integral {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (f g : Ω → ℝ)
(hf : Measurable f) (hg : Measurable g) (r t : ℝ) :
(∫ x, (if r ≤ f x then (1 : ℝ) else 0)*(if t ≤ g x then (1 : ℝ) else 0) ∂μ) =
(μ {x | r ≤ f x ∧ t ≤ g x}).toReal := by
have hs : MeasurableSet {x | r ≤ f x ∧ t ≤ g x} :=
(measurableSet_le measurable_const hf).inter (measurableSet_le measurable_const hg)
have he : (fun x => (if r ≤ f x then (1 : ℝ) else 0)*(if t ≤ g x then (1 : ℝ) else 0)) =
{x | r ≤ f x ∧ t ≤ g x}.indicator (fun _ => (1 : ℝ)) := by
funext x; simp only [Set.indicator,Set.mem_ofPred_eq]; split_ifs <;> simp_all
rw [he,integral_indicator hs]
simp [measureReal_def]
/-- The actual second-oriented reciprocal inherits rectangle discrepancy,
including both monotone pieces, cutoff jumps and reflected interval tests. -/
theorem samplingTestedSecond_pair_bound {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X Y : Ω → ℝ)
(hX : Measurable X) (hY : Measurable Y)
[NullSingletonClass (μ.map X)] [NullSingletonClass (μ.map Y)]
(hXs : ∀ᵐ x ∂μ.map X, x ∈ Set.Ico (0 : ℝ) 1)
(hYs : ∀ᵐ x ∂μ.map Y, x ∈ Set.Ico (0 : ℝ) 1)
(δ : ℝ) (hrect : ∀ a b c d : ℝ,
|(μ {x | X x ∈ Set.Icc a b ∧ Y x ∈ Set.Icc c d}).toReal -
(unitIntervalMeasure (Set.Icc a b)).toReal*(unitIntervalMeasure (Set.Icc c d)).toReal| ≤ δ)
(h α : ℝ) (G : Set ℝ) (a c s : ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hG : MeasurableSet G)
(hs : s ∈ Set.Ico (0 : ℝ) 1) :
|(∫ x, samplingTestedSecondWeight h α (Set.Icc a c) G s (X x)*
samplingTestedSecondWeight h α (Set.Icc a c) G s (Y x) ∂μ)-
samplingTestedSecondDensity h α (Set.Icc a c) G s^2| ≤
(4*δ)*(max 1 (1/(h*α)))^2 := by
let w := samplingTestedSecondWeight h α (Set.Icc a c) G s
let C := max 1 (1/(h*α))
have hw : Measurable w := (samplingTestedSecondWeight_measurable h α _ G measurableSet_Icc hG).comp
(measurable_const.prodMk measurable_id)
have hwb (t : ℝ) : 0 ≤ w t ∧ w t ≤ C :=
⟨(samplingTestedSecondWeight_bounds h α _ G s t hh hα).1,
(samplingTestedSecondWeight_bounds h α _ G s t hh hα).2.trans
(samplingWeight_bounds false h α _ t hh hα).2⟩
have : NullSingletonClass unitIntervalMeasure := by unfold unitIntervalMeasure; infer_instance
have hu : ∀ᵐ t ∂unitIntervalMeasure, t ∈ Set.Ico (0 : ℝ) 1 :=
self_mem_ae_restrict measurableSet_Ico
have hl (r t : ℝ) (hr : 0 < r) (_hrC : r ≤ C) (ht : 0 < t) (_htC : t ≤ C) :
|(μ {x | r ≤ w (X x) ∧ t ≤ w (Y x)}).toReal -
(unitIntervalMeasure.prod unitIntervalMeasure {p : ℝ × ℝ | r ≤ w p.1 ∧ t ≤ w p.2}).toReal| ≤ 4*δ := by
obtain ⟨A,B,hAB⟩ := samplingTestedSecond_level_exists h α G a c s r hh hh1 hα hs hr
obtain ⟨D,E,hDE⟩ := samplingTestedSecond_level_exists h α G a c s t hh hh1 hα hs ht
have hx := ae_of_ae_map hX.aemeasurable (hAB (μ.map X) hXs)
have hy := ae_of_ae_map hY.aemeasurable (hDE (μ.map Y) hYs)
have hpair : (∫ x, (if r ≤ w (X x) then (1 : ℝ) else 0)*
(if t ≤ w (Y x) then (1 : ℝ) else 0) ∂μ) =
∫ x, (∑ j, samplingClosedIndicator (A j) (B j) (X x))*
(∑ k, samplingClosedIndicator (D k) (E k) (Y x)) ∂μ := by
apply integral_congr_ae
filter_upwards [hx,hy] with x hx hy
rw [hx,hy]
have hmeanr := integral_congr_ae (hAB unitIntervalMeasure hu)
have hmeant := integral_congr_ae (hDE unitIntervalMeasure hu)
rw [← sampling_level_pair_integral μ (fun x => w (X x)) (fun x => w (Y x)) (hw.comp hX) (hw.comp hY) r t,hpair,
← sampling_level_pair_integral (unitIntervalMeasure.prod unitIntervalMeasure) (fun p => w p.1) (fun p => w p.2)
(hw.comp measurable_fst) (hw.comp measurable_snd) r t]
rw [integral_prod_mul (fun x => if r ≤ w x then (1 : ℝ) else 0)
(fun x => if t ≤ w x then (1 : ℝ) else 0)]
change |_- (∫ x, if r ≤ w x then (1 : ℝ) else 0 ∂unitIntervalMeasure)*
(∫ x, if t ≤ w x then (1 : ℝ) else 0 ∂unitIntervalMeasure)| ≤ _
rw [hmeanr,hmeant]
exact sampling_two_intervals_pair μ X Y hX hY δ hrect A B D E
have hc := sampling_layer_discrepancy μ (unitIntervalMeasure.prod unitIntervalMeasure)
(fun x => w (X x)) (fun x => w (Y x)) (fun p => w p.1) (fun p => w p.2)
(hw.comp hX) (hw.comp hY) (hw.comp measurable_fst) (hw.comp measurable_snd)
C (4*δ) (le_trans (by norm_num) (le_max_left _ _))
(fun x => hwb (X x)) (fun x => hwb (Y x)) (fun p => hwb p.1) (fun p => hwb p.2) hl
rw [integral_prod_mul w w] at hc
simpa only [w,C,samplingTestedSecondDensity,pow_two] using hc
end
end FiniteFactor
end OAI
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