| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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⟩ |
|
|
| / |
| 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 |
|
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| end OAI |
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