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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 | import OAI.Analysis.FiniteFactor.PeriodizedUniform
import OAI.Analysis.FiniteFactor.CrossedProduct.AngleLaw
import Mathlib.MeasureTheory.Integral.Bochner.Basic
namespace OAI
namespace FiniteFactor
open MeasureTheory ProbabilityTheory
open scoped ENNReal
noncomputable section
/-- Lower counterpart to periodized interval domination; the floor-length
subinterval contributes exact Haar measure. -/
theorem fract_interval_lower (a b : ℝ) (hab : a < b) :
ENNReal.ofReal (1-1/(b-a)) • unitIntervalMeasure ≤
(cond volume (Set.Ioc a b)).map Int.fract := by
let n : ℕ := Nat.floor (b-a)
have hpos : 0 < b-a := sub_pos.mpr hab
have hsub : Set.Ioc a (a+n) ⊆ Set.Ioc a b := by
apply Set.Ioc_subset_Ioc le_rfl
have hh : (n : ℝ) ≤ b-a := Nat.floor_le hpos.le
linarith
have hle : (n : ℝ≥0∞) • unitIntervalMeasure ≤
(volume.restrict (Set.Ioc a b)).map Int.fract := by
rw [← fract_integer_interval a n]
exact Measure.map_mono (Measure.restrict_mono hsub le_rfl) sourceFract_measurable
have hscalar : ENNReal.ofReal (1-1/(b-a)) ≤
(ENNReal.ofReal (b-a))⁻¹ * (n : ℝ≥0∞) := by
have hn : b-a < (n : ℝ)+1 := Nat.lt_floor_add_one (b-a)
have hreal : 1-1/(b-a) ≤ (b-a)⁻¹ * (n : ℝ) := by
apply (le_inv_mul_iff₀ hpos).mpr
field_simp
linarith
rw [← ENNReal.ofReal_inv_of_pos hpos, ← ENNReal.ofReal_natCast,
← ENNReal.ofReal_mul (inv_nonneg.mpr hpos.le)]
exact ENNReal.ofReal_le_ofReal hreal
unfold ProbabilityTheory.cond
rw [Measure.map_smul _ sourceFract_measurable.aemeasurable, Real.volume_Ioc]
calc
_ ≤ ((ENNReal.ofReal (b-a))⁻¹ * (n : ℝ≥0∞)) • unitIntervalMeasure := by gcongr
_ = (ENNReal.ofReal (b-a))⁻¹ • ((n : ℝ≥0∞) • unitIntervalMeasure) :=
(smul_smul _ _ _).symm
_ ≤ _ := by gcongr
/-- Two-sided near-uniformity for the actual single-phase law, on every
Borel set. The error is exactly 1/(n times the source interval length). -/
theorem sourceAngle_phase_sandwich (i : SourceCoordinate) (n : ℝ) (hn : 0 < n) :
ENNReal.ofReal (1-1/(n*sourceAngleUnit i)) • unitIntervalMeasure ≤
(sourceAngleMarginal i).map (fun t => Int.fract (n*t)) ∧
(sourceAngleMarginal i).map (fun t => Int.fract (n*t)) ≤
ENNReal.ofReal (1+1/(n*sourceAngleUnit i)) • unitIntervalMeasure := by
have ha := sourceAngleUnit_pos i
have he : (sourceAngleMarginal i).map (fun t => Int.fract (n*t)) =
(cond volume (Set.Ioc (n*sourceAngleUnit i) (n*(2*sourceAngleUnit i)))).map Int.fract := by
rw [← cond_interval_mul_pos (sourceAngleUnit i) (2*sourceAngleUnit i) n (by linarith) hn,
Measure.map_map sourceFract_measurable (by fun_prop)]
rw [sourceAngleMarginal, sourceAngleInterval, cond_closed_interval_eq]
rfl
rw [he]
have hpos : n*sourceAngleUnit i < n*(2*sourceAngleUnit i) := by nlinarith
have hlen : n*(2*sourceAngleUnit i)-n*sourceAngleUnit i = n*sourceAngleUnit i := by ring
constructor
· simpa only [hlen] using fract_interval_lower _ _ hpos
· simpa only [hlen] using fract_interval_domination _ _ hpos
/-- Nonnegative tests inherit the exact relative error from a measure sandwich. -/
theorem integral_error_of_measure_sandwich {α : Type*} [MeasurableSpace α]
(μ ν : Measure α) (ε : ℝ) (hε : 0 ≤ ε) (hε1 : ε ≤ 1)
(hlo : ENNReal.ofReal (1-ε) • μ ≤ ν)
(hhi : ν ≤ ENNReal.ofReal (1+ε) • μ)
(g : α → ℝ) (hg : ∀ x, 0 ≤ g x) (hgi : Integrable g μ) :
Integrable g ν ∧ |(∫ x, g x ∂ν) - ∫ x, g x ∂μ| ≤ ε * ∫ x, g x ∂μ := by
have hgui := hgi.smul_measure (c := ENNReal.ofReal (1+ε)) ENNReal.ofReal_ne_top
have hgν := hgui.mono_measure hhi
refine ⟨hgν, ?_⟩
have hl := integral_mono_measure hlo (Filter.Eventually.of_forall hg) hgν
have hu := integral_mono_measure hhi (Filter.Eventually.of_forall hg) hgui
rw [integral_smul_measure, ENNReal.toReal_ofReal (by linarith), smul_eq_mul] at hl
rw [integral_smul_measure, ENNReal.toReal_ofReal (by linarith), smul_eq_mul] at hu
exact abs_le.mpr ⟨by nlinarith, by nlinarith⟩
/-- The source's single-phase test estimate for all integrable nonnegative
weights, not merely for rectangle indicators. -/
theorem sourceAngle_single_phase_test (i : SourceCoordinate) (n : ℝ) (hn : 0 < n)
(hlen : 1 ≤ n*sourceAngleUnit i) (g : ℝ → ℝ)
(hg : ∀ x, 0 ≤ g x) (hgm : Measurable g) (hgi : Integrable g unitIntervalMeasure) :
|(∫ t, g (Int.fract (n*t)) ∂sourceAngleMarginal i) -
∫ t, g t ∂unitIntervalMeasure| ≤
(1/(n*sourceAngleUnit i)) * ∫ t, g t ∂unitIntervalMeasure := by
have hp := sourceAngle_phase_sandwich i n hn
have he0 : 0 ≤ 1/(n*sourceAngleUnit i) := by positivity
have he1 : 1/(n*sourceAngleUnit i) ≤ 1 := by
exact (one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 1) hlen).trans_eq (by norm_num)
have hh := (integral_error_of_measure_sandwich unitIntervalMeasure _ _ he0 he1 hp.1 hp.2 g hg hgi).2
have hphase : Measurable (fun t : ℝ => Int.fract (n*t)) :=
sourceFract_measurable.comp (measurable_const.mul measurable_id)
rw [integral_map hphase.aemeasurable hgm.aestronglyMeasurable] at hh
exact hh
end
end FiniteFactor
end OAI
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