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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 | import OAI.Analysis.FiniteFactor.CrossedProduct.Rotation
import Mathlib.Probability.ConditionalProbability
namespace OAI
namespace FiniteFactor
open MeasureTheory ProbabilityTheory
open scoped ENNReal
noncomputable section
theorem sourceFract_measurable : Measurable (Int.fract : ℝ → ℝ) := by
have hh := sourceCircleRepresentative_measurable.comp sourceCircle_mk_preserving.measurable
simpa only [Function.comp_def, sourceCircleRepresentative_coe] using hh
/-- Every translated interval of length one maps to the same representative
Haar measure under fractional part. -/
theorem fract_unit_interval (a : ℝ) :
(volume.restrict (Set.Ioc a (a+1))).map Int.fract = unitIntervalMeasure := by
have hh := sourceCircleRepresentative_preserving.comp (AddCircle.measurePreserving_mk 1 a)
have he : sourceCircleRepresentative ∘ ((↑) : ℝ → AddCircle (1 : ℝ)) = Int.fract := by
funext x
exact sourceCircleRepresentative_coe x
simpa only [he] using hh.map_eq
theorem fract_integer_interval (a : ℝ) (n : ℕ) :
(volume.restrict (Set.Ioc a (a+n))).map Int.fract = (n : ℝ≥0∞) • unitIntervalMeasure := by
induction n with
| zero => simp
| succ n ih =>
have hu : Set.Ioc a (a+((n+1 : ℕ) : ℝ)) =
Set.Ioc a (a+n) ∪ Set.Ioc (a+n) (a+n+1) := by
rw [Set.Ioc_union_Ioc_eq_Ioc (by have := Nat.cast_nonneg (α := ℝ) n; linarith) (by linarith)]
congr 1
push_cast
ring
rw [hu, Measure.restrict_union (Set.Ioc_disjoint_Ioc_of_le le_rfl) measurableSet_Ioc,
Measure.map_add, ih, fract_unit_interval]
· simp [add_smul]
· exact sourceFract_measurable
/-- Periodizing normalized interval measure has density at most 1+1/length.
The bound is expressed as domination of measures on every measurable set. -/
theorem fract_interval_domination (a b : ℝ) (hab : a < b) :
(cond volume (Set.Ioc a b)).map Int.fract ≤
ENNReal.ofReal (1+1/(b-a)) • unitIntervalMeasure := by
let n : ℕ := Nat.ceil (b-a)
have hsub : Set.Ioc a b ⊆ Set.Ioc a (a+n) := by
apply Set.Ioc_subset_Ioc le_rfl
have hh : b-a ≤ (n : ℝ) := Nat.le_ceil _
linarith
have hle : (volume.restrict (Set.Ioc a b)).map Int.fract ≤
(n : ℝ≥0∞) • unitIntervalMeasure := by
rw [← fract_integer_interval a n]
exact Measure.map_mono (Measure.restrict_mono hsub le_rfl) sourceFract_measurable
have hscalar : (ENNReal.ofReal (b-a))⁻¹ * (n : ℝ≥0∞) ≤
ENNReal.ofReal (1+1/(b-a)) := by
have hpos : 0 < b-a := sub_pos.mpr hab
have hn : (n : ℝ) < b-a+1 := Nat.ceil_lt_add_one hpos.le
have hreal : (b-a)⁻¹ * (n : ℝ) ≤ 1+1/(b-a) := by
apply (mul_le_mul_of_nonneg_left hn.le (inv_nonneg.mpr hpos.le)).trans_eq
field_simp
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 _ _ _
_ ≤ _ := by gcongr
/-- Positive scaling carries normalized interval measure to the normalized
scaled interval; this identifies the actual phase law before periodization. -/
theorem cond_interval_mul_pos (a b c : ℝ) (hab : a < b) (hc : 0 < c) :
(cond volume (Set.Ioc a b)).map (fun t => c*t) =
cond volume (Set.Ioc (c*a) (c*b)) := by
have hp : (fun t : ℝ => c*t) ⁻¹' Set.Ioc (c*a) (c*b) = Set.Ioc a b := by
ext t
simp only [Set.mem_preimage, Set.mem_Ioc, mul_lt_mul_iff_right₀ hc, mul_le_mul_iff_right₀ hc]
have hlen : 0 < b-a := sub_pos.mpr hab
have hclen : 0 < c*b-c*a := by nlinarith
unfold ProbabilityTheory.cond
rw [Measure.map_smul _ (by fun_prop), Real.volume_Ioc, Real.volume_Ioc, ← hp,
← Measure.restrict_map (f := fun t : ℝ => c*t) (by fun_prop) measurableSet_Ioc,
Real.map_volume_mul_left hc.ne', Measure.restrict_smul, smul_smul]
congr 1
rw [abs_of_pos (inv_pos.mpr hc), ← ENNReal.ofReal_inv_of_pos hlen,
← ENNReal.ofReal_mul (inv_nonneg.mpr hlen.le), ← ENNReal.ofReal_inv_of_pos hclen]
congr 1
field_simp
theorem cond_closed_interval_eq (a b : ℝ) :
cond volume (Set.Icc a b) = cond volume (Set.Ioc a b) := by
unfold ProbabilityTheory.cond
rw [Real.volume_Icc, Real.volume_Ioc, ← restrict_Ioc_eq_restrict_Icc]
end
end FiniteFactor
end OAI
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