| import OAI.Analysis.FiniteFactor.OrbitSampling.CutoffMass |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical ENNReal Topology |
| noncomputable section |
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| / |
| 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] |
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| / |
| 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] |
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| / |
| 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] |
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| end |
| end FiniteFactor |
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| end OAI |
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