| import OAI.Analysis.FiniteFactor.Sampling.Weights |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| noncomputable section |
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| def samplingTestedFirstWeight (b : Bool) (h α : ℝ) (E G : Set ℝ) (s t : ℝ) : ℝ := by |
| classical |
| exact if s ∈ E ∧ Int.fract (s+t) ∈ G then samplingWeight b h α s t else 0 |
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| def samplingTestedFirstDensity (b : Bool) (h α : ℝ) (E G : Set ℝ) (s : ℝ) : ℝ := |
| ∫ t, samplingTestedFirstWeight b h α E G s t ∂unitIntervalMeasure |
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| theorem samplingTestedFirstWeight_measurable (b : Bool) (h α : ℝ) (E G : Set ℝ) |
| (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| Measurable (fun p : ℝ × ℝ => samplingTestedFirstWeight b h α E G p.1 p.2) := by |
| unfold samplingTestedFirstWeight |
| exact Measurable.ite ((hE.preimage measurable_fst).inter |
| (hG.preimage (sourceFract_measurable.comp (measurable_fst.add measurable_snd)))) |
| (samplingWeight_measurable b h α) measurable_const |
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| theorem samplingTestedFirstWeight_bounds (b : Bool) (h α : ℝ) (E G : Set ℝ) (s t : ℝ) |
| (hh : 0 < h) (hα : 0 < α) : |
| 0 ≤ samplingTestedFirstWeight b h α E G s t ∧ |
| samplingTestedFirstWeight b h α E G s t ≤ samplingWeight b h α s t := by |
| have hn := (samplingWeight_bounds b h α s t hh hα).1 |
| unfold samplingTestedFirstWeight |
| split_ifs <;> simp [hn] |
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| theorem samplingWeight_integrable (b : Bool) (h α s : ℝ) (hh : 0 < h) (hα : 0 < α) : |
| Integrable (samplingWeight b h α s) unitIntervalMeasure := by |
| apply (integrable_const (max 1 (1/(h*α)))).mono' |
| ((samplingWeight_measurable b h α).comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable |
| exact Eventually.of_forall (fun t => by |
| change ‖samplingWeight b h α s t‖ ≤ _ |
| rw [Real.norm_eq_abs,abs_of_nonneg (samplingWeight_bounds b h α s t hh hα).1] |
| exact (samplingWeight_bounds b h α s t hh hα).2) |
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| theorem samplingTestedFirstDensity_measurable (b : Bool) (h α : ℝ) (E G : Set ℝ) |
| (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| Measurable (samplingTestedFirstDensity b h α E G) := |
| (samplingTestedFirstWeight_measurable b h α E G hE hG).stronglyMeasurable.integral_prod_right.measurable |
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| / |
| theorem samplingTestedFirstDensity_bounds (b : Bool) (h α : ℝ) (E G : Set ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hE : MeasurableSet E) (hG : MeasurableSet G) |
| (s : ℝ) (hs : s ∈ Set.Ico (0 : ℝ) 1) : |
| 0 ≤ samplingTestedFirstDensity b h α E G s ∧ |
| samplingTestedFirstDensity b h α E G s ≤ 1 := by |
| have hm : Measurable (samplingTestedFirstWeight b h α E G s) := (samplingTestedFirstWeight_measurable b h α E G hE hG).comp |
| (measurable_const.prodMk measurable_id) |
| have hi := (samplingWeight_integrable b h α s hh hα).mono' hm.aestronglyMeasurable |
| (Eventually.of_forall (fun t => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (samplingTestedFirstWeight_bounds b h α E G s t hh hα).1] |
| exact (samplingTestedFirstWeight_bounds b h α E G s t hh hα).2)) |
| refine ⟨integral_nonneg (fun t => (samplingTestedFirstWeight_bounds b h α E G s t hh hα).1), ?_⟩ |
| calc |
| _ ≤ ∫ t, samplingWeight b h α s t ∂unitIntervalMeasure := |
| integral_mono_ae hi (samplingWeight_integrable b h α s hh hα) |
| (Eventually.of_forall (fun t => (samplingTestedFirstWeight_bounds b h α E G s t hh hα).2)) |
| _ ≤ 1 := by |
| rw [samplingWeight_first_integral b h α s hh hh1 hα hs] |
| cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] |
| · split_ifs <;> norm_num |
| · linarith [hs.1] |
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|
| / |
| triangular and diagonal coordinates, uniformly in the measurable tests. -/ |
| theorem samplingTestedFirstDensity_square (b : Bool) (h α : ℝ) (E G : Set ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) (hE : MeasurableSet E) (hG : MeasurableSet G) : |
| (∫ s, samplingTestedFirstDensity b h α E G s ^ 2 ∂unitIntervalMeasure) ≤ 1 := by |
| have hb : ∀ᵐ s ∂unitIntervalMeasure, 0 ≤ samplingTestedFirstDensity b h α E G s ∧ |
| samplingTestedFirstDensity b h α E G s ≤ 1 := |
| (ae_restrict_mem measurableSet_Ico).mono (fun s hs => |
| samplingTestedFirstDensity_bounds b h α E G hh hh1 hα hE hG s hs) |
| have hb2 := hb.mono (fun s hs => show samplingTestedFirstDensity b h α E G s ^ 2 ≤ 1 by nlinarith) |
| have hi : Integrable (fun s => samplingTestedFirstDensity b h α E G s ^ 2) unitIntervalMeasure := |
| (integrable_const (1 : ℝ)).mono' |
| ((samplingTestedFirstDensity_measurable b h α E G hE hG).pow_const 2).aestronglyMeasurable |
| (hb2.mono (fun s hs => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg (samplingTestedFirstDensity b h α E G s))] |
| exact hs)) |
| simpa using integral_mono_ae hi (integrable_const (1 : ℝ)) hb2 |
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| end |
| end FiniteFactor |
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| end OAI |
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