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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKGap.Model
namespace OAI
section
noncomputable section
namespace SKGap
open MeasureTheory Real Set
lemma interval_product_integrable {Ω : Type*} [TopologicalSpace Ω] [MeasurableSpace Ω]
[BorelSpace Ω] [SecondCountableTopology Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
{B : ℝ × Ω → ℝ} (hB : Continuous B) {D : ℝ}
(hb : ∀ t ∈ Icc (0:ℝ) 1, ∀ g, |B (t,g)| ≤ D) :
Integrable B ((volume.restrict (Ioc (0:ℝ) 1)).prod μ) := by
apply Integrable.of_bound hB.aestronglyMeasurable D
apply (Measure.ae_prod_iff_ae_ae (measurableSet_le hB.norm.measurable measurable_const)).mpr
filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
exact ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using hb t ⟨ht.1.le,ht.2⟩ g)
lemma bounded_interval_mean {Ω : Type*} [TopologicalSpace Ω] [MeasurableSpace Ω]
[BorelSpace Ω] [SecondCountableTopology Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
{B : ℝ × Ω → ℝ} (hB : Continuous B) {D c ε : ℝ}
(hb : ∀ t ∈ Icc (0:ℝ) 1, ∀ g, |B (t,g)| ≤ D)
(hm : ∀ t ∈ Icc (0:ℝ) 1, |(∫ g, B (t,g) ∂μ)-t*c| ≤ ε) :
Integrable (fun g => ∫ t in (0:ℝ)..1, B (t,g)) μ ∧
|(∫ g, (∫ t in (0:ℝ)..1, B (t,g)) ∂μ)-c/2| ≤ ε := by
have hi := interval_product_integrable (μ := μ) hB hb
have hi' : Integrable (Function.uncurry (fun t g => B (t,g)))
((volume.restrict (uIoc (0:ℝ) 1)).prod μ) := by
have he : Function.uncurry (fun t g => B (t,g)) = B := by
funext x; rcases x with ⟨t,g⟩; rfl
rw [he,uIoc_of_le (by norm_num : (0:ℝ)≤1)]
exact hi
constructor
· simpa only [intervalIntegral.integral_of_le (by norm_num : (0:ℝ)≤1)] using hi.integral_prod_right
· rw [← intervalIntegral_integral_swap hi']
have hit : IntervalIntegrable (fun t => ∫ g, B (t,g) ∂μ) volume 0 1 := by
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le (by norm_num : (0:ℝ)≤1)]
exact hi.integral_prod_left
have hid : IntervalIntegrable (fun t : ℝ => t*c) volume 0 1 := (continuous_id.mul_const c).intervalIntegrable 0 1
have hc : (∫ t in (0:ℝ)..1, t*c)=c/2 := by rw [intervalIntegral.integral_mul_const,integral_id]; ring
rw [← hc,← intervalIntegral.integral_sub hit hid]
have hh := intervalIntegral.norm_integral_le_of_norm_le_const (a := (0:ℝ)) (b := 1)
(f := fun t => (∫ g, B (t,g) ∂μ)-t*c) (C := ε) (fun t ht => by
have ht' : t ∈ Ioc (0:ℝ) 1 := by simpa only [uIoc_of_le (by norm_num : (0:ℝ)≤1)] using ht
simpa only [Real.norm_eq_abs] using hm t ⟨ht'.1.le,ht'.2⟩)
simpa using hh
end SKGap
end
end
end OAI
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