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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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | 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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