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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 | import OAI.Analysis.FiniteFactor.OrbitSampling.CutoffMass
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical ENNReal Topology
noncomputable section
/-- The actual triangular phase weight is precisely the order indicator
of the rotated endpoint, on the Haar representatives. -/
theorem samplingWeight_triangular_order (h α s t : ℝ)
(hs : s ∈ Set.Ico (0 : ℝ) 1) (ht : t ∈ Set.Ico (0 : ℝ) 1) :
samplingWeight true h α s t = if s < Int.fract (s+t) then 1 else 0 := by
by_cases hst : s+t < 1
· have hf : Int.fract (s+t) = s+t := Int.fract_eq_self.mpr ⟨by linarith [hs.1,ht.1],hst⟩
simp only [samplingWeight,ite_true,hf]
have he : (0 < t ∧ t < 1-s) ↔ s < s+t := by
constructor
· intro h; linarith [h.1]
· intro h; constructor <;> linarith
simp only [he]
· have hf : ⌊s+t⌋ = (1 : ℤ) := Int.floor_eq_iff.mpr ⟨by norm_num; linarith,by norm_num; linarith [hs.2,ht.2]⟩
have hn : ¬ (0 < t ∧ t < 1-s) := by intro h; linarith [h.2]
have ho : ¬ s < Int.fract (s+t) := by rw [Int.fract,hf]; norm_num; linarith [ht.2]
simp only [samplingWeight,ite_true,ite_eq_right hn,ite_eq_right ho]
/-- The first tested triangular density is its actual vertical section of
Lebesgue measure on 0≤x<y<1. No triangular marginal identity is assumed. -/
theorem samplingTestedFirstDensity_triangular_section (h α s : ℝ) (E G : Set ℝ)
(hG : MeasurableSet G) (hs : s ∈ Set.Ico (0 : ℝ) 1) :
samplingTestedFirstDensity true h α E G s =
∫ y, (if s ∈ E ∧ y ∈ G ∧ s < y then (1 : ℝ) else 0) ∂unitIntervalMeasure := by
let F := fun y : ℝ => if s ∈ E ∧ y ∈ G ∧ s < y then (1 : ℝ) else 0
have hm : Measurable F := by
apply Measurable.ite _ measurable_const measurable_const
by_cases htest : s ∈ E
· simp only [htest,true_and]
change MeasurableSet (G ∩ {y : ℝ | s < y})
exact hG.inter (measurableSet_lt measurable_const measurable_id)
· simp [htest]
have hp : MeasurePreserving (fun t : ℝ => Int.fract (s+t)) unitIntervalMeasure unitIntervalMeasure := by
simpa only [add_comm] using unitInterval_rotate_preserving s
have he : samplingTestedFirstWeight true h α E G s =ᵐ[unitIntervalMeasure]
(fun t => F (Int.fract (s+t))) := by
filter_upwards [ae_restrict_mem measurableSet_Ico] with t ht
rw [samplingTestedFirstWeight,samplingWeight_triangular_order h α s t hs ht]
simp only [F]
split_ifs <;> simp_all
rw [samplingTestedFirstDensity,integral_congr_ae he]
rw [← integral_map hp.measurable.aemeasurable hm.aestronglyMeasurable,hp.map_eq]
/-- The two-dimensional tested source integral equals the genuine triangular
restriction of product Lebesgue measure. -/
theorem samplingTestedFirstDensity_triangular_mass (h α : ℝ) (E G : Set ℝ)
(hE : MeasurableSet E) (hG : MeasurableSet G) :
(∫ s, samplingTestedFirstDensity true h α E G s ∂unitIntervalMeasure) =
((unitIntervalMeasure.prod unitIntervalMeasure)
{p : ℝ × ℝ | p.1 ∈ E ∧ p.2 ∈ G ∧ p.1 < p.2}).toReal := by
let A := {p : ℝ × ℝ | p.1 ∈ E ∧ p.2 ∈ G ∧ p.1 < p.2}
have hA : MeasurableSet A := (hE.preimage measurable_fst).inter
((hG.preimage measurable_snd).inter (measurableSet_lt measurable_fst measurable_snd))
have hm : Measurable (A.indicator (fun _ => (1 : ℝ))) := measurable_const.indicator hA
have hi : Integrable (A.indicator (fun _ => (1 : ℝ))) (unitIntervalMeasure.prod unitIntervalMeasure) :=
(integrable_const (1 : ℝ)).indicator hA
calc
_ = ∫ s, ∫ y, A.indicator (fun _ => (1 : ℝ)) (s,y) ∂unitIntervalMeasure ∂unitIntervalMeasure := by
apply integral_congr_ae
filter_upwards [ae_restrict_mem measurableSet_Ico] with s hs
simpa only [A,Set.indicator_apply,Set.mem_ofPred_eq] using
samplingTestedFirstDensity_triangular_section h α s E G hG hs
_ = ∫ p, A.indicator (fun _ => (1 : ℝ)) p ∂unitIntervalMeasure.prod unitIntervalMeasure :=
(integral_prod _ hi).symm
_ = _ := by rw [integral_indicator hA]; simp [measureReal_def,A]
end
end FiniteFactor
end OAI
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