| import OAI.Analysis.FiniteFactor.Sampling.IntervalError |
| import OAI.Analysis.FiniteFactor.CrossedProduct.PhaseMarginal |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory |
| open scoped ENNReal |
| noncomputable section |
|
|
| def samplingBoundary (a b δ : ℝ) : Set ℝ := |
| (Set.Icc δ (1-δ))ᶜ ∪ Set.Icc (a-δ) (a+δ) ∪ Set.Icc (b-δ) (b+δ) |
|
|
| theorem samplingBoundary_measurable (a b δ : ℝ) : MeasurableSet (samplingBoundary a b δ) := |
| (measurableSet_Icc.compl.union measurableSet_Icc).union measurableSet_Icc |
|
|
| theorem samplingBoundary_measure (a b δ : ℝ) (hδ : 0 ≤ δ) : |
| unitIntervalMeasure (samplingBoundary a b δ) ≤ ENNReal.ofReal (6*δ) := by |
| have hband (a : ℝ) : unitIntervalMeasure (Set.Icc (a-δ) (a+δ)) ≤ ENNReal.ofReal (2*δ) := by |
| calc |
| _ ≤ volume (Set.Icc (a-δ) (a+δ)) := Measure.restrict_le_self _ |
| _ = _ := by rw [Real.volume_Icc]; congr 1; ring |
| calc |
| _ ≤ unitIntervalMeasure ((Set.Icc δ (1-δ))ᶜ ∪ Set.Icc (a-δ) (a+δ)) + |
| unitIntervalMeasure (Set.Icc (b-δ) (b+δ)) := measure_union_le _ _ |
| _ ≤ unitIntervalMeasure (Set.Icc δ (1-δ))ᶜ + |
| unitIntervalMeasure (Set.Icc (a-δ) (a+δ)) + |
| unitIntervalMeasure (Set.Icc (b-δ) (b+δ)) := by gcongr; exact measure_union_le _ _ |
| _ ≤ ENNReal.ofReal (2*δ) + ENNReal.ofReal (2*δ) + ENNReal.ofReal (2*δ) := by |
| gcongr |
| · exact unitInterval_middle_compl δ hδ |
| · exact hband a |
| · exact hband b |
| _ = _ := by rw [← ENNReal.ofReal_add (by positivity) (by positivity), |
| ← ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring |
|
|
| theorem samplingSmooth_indicator_error_le_one (H : ℕ) (hH : 0 < H) (E : Set ℝ) (t : ℝ) : |
| |samplingSmooth H E t - E.indicator (fun _ => (1 : ℝ)) t| ≤ 1 := by |
| have h := samplingSmooth_bounds H hH E t |
| by_cases ht : t ∈ E |
| · rw [Set.indicator_of_mem ht] |
| exact abs_le.mpr ⟨by linarith [h.1], by linarith [h.2]⟩ |
| · rw [Set.indicator_of_notMem ht, sub_zero, abs_of_nonneg h.1] |
| exact h.2 |
|
|
| /-- The interval smoothing error integrated under any actual phase law with |
| density bounded by two. Only one-dimensional domination is used. -/ |
| theorem samplingSmooth_expected_error (ν : Measure ℝ) [IsProbabilityMeasure ν] |
| (hν : ν ≤ (2 : ℝ≥0∞) • unitIntervalMeasure) (H : ℕ) (hH : 0 < H) |
| (a b δ : ℝ) (hδ : 0 < δ) : |
| (∫ t, |samplingSmooth H (Set.Icc a b) t - |
| (Set.Icc a b).indicator (fun _ => (1 : ℝ)) t| ∂ν) ≤ |
| 12*δ + 1/(4*(H : ℝ)*δ^2) := by |
| let E := Set.Icc a b |
| let B := samplingBoundary a b δ |
| let C := 1/(4*(H : ℝ)*δ^2) |
| let f := fun t => |samplingSmooth H E t - E.indicator (fun _ => (1 : ℝ)) t| |
| have hC : 0 ≤ C := by dsimp [C]; positivity |
| have hBm : MeasurableSet B := samplingBoundary_measurable a b δ |
| have hfm : Measurable f := by |
| have hi : Measurable (E.indicator (fun _ => (1 : ℝ))) := measurable_const.indicator measurableSet_Icc |
| simpa only [Real.norm_eq_abs, Pi.sub_apply, f] using ((samplingSmooth_continuous H E).measurable.sub hi).norm |
| have hfi : Integrable f ν := (integrable_const (1 : ℝ)).mono' hfm.aestronglyMeasurable |
| (Filter.Eventually.of_forall (fun t => by |
| rw [Real.norm_eq_abs, abs_of_nonneg (abs_nonneg _)] |
| exact samplingSmooth_indicator_error_le_one H hH E t)) |
| have hle (t : ℝ) : f t ≤ B.indicator (fun _ => (1 : ℝ)) t + C := by |
| by_cases ht : t ∈ B |
| · rw [Set.indicator_of_mem ht] |
| exact (samplingSmooth_indicator_error_le_one H hH E t).trans (by linarith) |
| · rw [Set.indicator_of_notMem ht, zero_add] |
| have hn := not_or.mp ht |
| have hn' := not_or.mp hn.1 |
| have hmid : t ∈ Set.Icc δ (1-δ) := by simpa only [Set.mem_compl_iff, not_not] using hn'.1 |
| have ha : δ < |t-a| := by |
| by_contra hh |
| have hab := abs_le.mp (le_of_not_gt hh) |
| exact hn'.2 ⟨by linarith [hab.1], by linarith [hab.2]⟩ |
| have hb : δ < |t-b| := by |
| by_contra hh |
| have hab := abs_le.mp (le_of_not_gt hh) |
| exact hn.2 ⟨by linarith [hab.1], by linarith [hab.2]⟩ |
| exact samplingSmooth_interval_error H hH a b δ t hδ hmid.1 hmid.2 ha hb |
| have hBμ : ν B ≤ ENNReal.ofReal (12*δ) := by |
| calc |
| _ ≤ ((2 : ℝ≥0∞) • unitIntervalMeasure) B := hν B |
| _ = 2 * unitIntervalMeasure B := rfl |
| _ ≤ 2 * ENNReal.ofReal (6*δ) := by gcongr; exact samplingBoundary_measure a b δ hδ.le |
| _ = _ := by rw [show (2 : ℝ≥0∞) = ENNReal.ofReal 2 by norm_num, |
| ← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 2)]; congr 1; ring |
| have hBreal : (ν B).toReal ≤ 12*δ := by |
| have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top hBμ |
| rwa [ENNReal.toReal_ofReal (by positivity)] at hh |
| have hh := integral_mono_ae hfi (((integrable_const (1 : ℝ)).indicator hBm).add (integrable_const C)) |
| (Filter.Eventually.of_forall hle) |
| simp only [Pi.add_apply] at hh |
| rw [integral_add ((integrable_const (1 : ℝ)).indicator hBm) (integrable_const C), integral_indicator hBm] at hh |
| simp [measureReal_def] at hh |
| change (∫ t, f t ∂ν) ≤ 12*δ+C |
| linarith |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
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