| import Mathlib |
| import OAI.Probability.SKGap.Model |
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| namespace OAI |
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| 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) |
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| 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 |
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| end OAI |
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