| import OAI.Analysis.FiniteFactor.Sampling.SecondDensity |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal |
| noncomputable section |
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| / |
| def samplingTestedSecondWeight (h α : ℝ) (E G : Set ℝ) (s t : ℝ) : ℝ := by |
| classical |
| exact if Int.fract (s-t) ∈ E ∧ s ∈ G then samplingSecondWeight h α s t else 0 |
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| def samplingTestedSecondDensity (h α : ℝ) (E G : Set ℝ) (s : ℝ) : ℝ := |
| ∫ t, samplingTestedSecondWeight h α E G s t ∂unitIntervalMeasure |
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| theorem samplingTestedSecondWeight_measurable (h α : ℝ) (E G : Set ℝ) |
| (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| Measurable (fun p : ℝ × ℝ => samplingTestedSecondWeight h α E G p.1 p.2) := by |
| unfold samplingTestedSecondWeight |
| exact Measurable.ite ((hE.preimage (sourceFract_measurable.comp |
| (measurable_fst.sub measurable_snd))).inter (hG.preimage measurable_fst)) |
| (samplingSecondWeight_measurable h α) measurable_const |
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| theorem samplingTestedSecondWeight_bounds (h α : ℝ) (E G : Set ℝ) (s t : ℝ) |
| (hh : 0 < h) (hα : 0 < α) : |
| 0 ≤ samplingTestedSecondWeight h α E G s t ∧ |
| samplingTestedSecondWeight h α E G s t ≤ samplingSecondWeight h α s t := by |
| have hn : 0 ≤ samplingSecondWeight h α s t := (samplingWeight_bounds false h α _ t hh hα).1 |
| unfold samplingTestedSecondWeight |
| split_ifs <;> simp [hn] |
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| theorem samplingTestedSecondDensity_measurable (h α : ℝ) (E G : Set ℝ) |
| (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| Measurable (samplingTestedSecondDensity h α E G) := |
| (samplingTestedSecondWeight_measurable h α E G hE hG).stronglyMeasurable.integral_prod_right.measurable |
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| theorem samplingTestedSecondDensity_bounds (h α : ℝ) (E G : Set ℝ) |
| (hh : 0 < h) (hα : 0 < α) (hE : MeasurableSet E) (hG : MeasurableSet G) (s : ℝ) : |
| 0 ≤ samplingTestedSecondDensity h α E G s ∧ |
| samplingTestedSecondDensity h α E G s ≤ samplingSecondDensity h α s := by |
| have hm : Measurable (samplingTestedSecondWeight h α E G s) := |
| (samplingTestedSecondWeight_measurable h α E G hE hG).comp (measurable_const.prodMk measurable_id) |
| have hi := (samplingSecondWeight_integrable h α s hh hα).mono' hm.aestronglyMeasurable |
| (Eventually.of_forall (fun t => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (samplingTestedSecondWeight_bounds h α E G s t hh hα).1] |
| exact (samplingTestedSecondWeight_bounds h α E G s t hh hα).2)) |
| exact ⟨integral_nonneg (fun t => (samplingTestedSecondWeight_bounds h α E G s t hh hα).1), |
| integral_mono_ae hi (samplingSecondWeight_integrable h α s hh hα) |
| (Eventually.of_forall (fun t => (samplingTestedSecondWeight_bounds h α E G s t hh hα).2))⟩ |
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| theorem samplingTestedSecondDensity_square (h α : ℝ) (E G : Set ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| (∀ᵐ s ∂unitIntervalMeasure, 0 ≤ samplingTestedSecondDensity h α E G s ∧ |
| samplingTestedSecondDensity h α E G s ≤ 1/(1-h)) ∧ |
| (∫ s, samplingTestedSecondDensity h α E G s^2 ∂unitIntervalMeasure) ≤ 1+h^2/(1-h)^2 := by |
| let f := samplingTestedSecondDensity h α E G |
| let q := samplingIdealSecond h |
| let C := 1/(1-h) |
| have hC : 0 ≤ C := by dsimp [C]; positivity |
| have hf := samplingTestedSecondDensity_measurable h α E G hE hG |
| have hq := samplingIdealSecond_measurable h hh |
| have hnon (s : ℝ) : 0 ≤ f s := (samplingTestedSecondDensity_bounds h α E G hh hα hE hG s).1 |
| have hle : ∀ᵐ s ∂unitIntervalMeasure, f s ≤ q s := |
| (samplingSecondDensity_le_ideal h α hh hh1 hα).mono (fun s hs => |
| (samplingTestedSecondDensity_bounds h α E G hh hα hE hG s).2.trans hs) |
| have hqb (s : ℝ) : 0 ≤ q s ∧ q s ≤ C := (samplingIdealSecond_bounds h s hh hh1).imp_right And.left |
| have hq2 (s : ℝ) : q s^2 ≤ C^2 := by nlinarith [(hqb s).1,(hqb s).2] |
| have hf2 : ∀ᵐ s ∂unitIntervalMeasure, f s^2 ≤ q s^2 := hle.mono (fun s hs => by nlinarith [hnon s,(hqb s).1]) |
| have hfi : Integrable (fun s => f s^2) unitIntervalMeasure := |
| (integrable_const (C^2)).mono' (hf.pow_const 2).aestronglyMeasurable (hf2.mono (fun s hs => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| exact hs.trans (hq2 s))) |
| have hqi : Integrable (fun s => q s^2) unitIntervalMeasure := |
| (integrable_const (C^2)).mono' (hq.pow_const 2).aestronglyMeasurable (Eventually.of_forall (fun s => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| exact hq2 s)) |
| exact ⟨hle.mono (fun s hs => ⟨hnon s,hs.trans (hqb s).2⟩), |
| (integral_mono_ae hfi hqi hf2).trans (samplingIdealSecond_square h hh hh1).2⟩ |
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| end |
| end FiniteFactor |
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| end OAI |
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