| import OAI.Analysis.FiniteFactor.Sampling.SecondLevelAE |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical ENNReal |
| noncomputable section |
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| 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] |
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| / |
| 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 |
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| end |
| end FiniteFactor |
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| end OAI |
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