| import Mathlib |
| import OAI.Probability.Ballisticity.Model |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology Classical |
| namespace DirectionalTransience |
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| lemma kernel_bounded_integral {Ω G : Type*} [MeasurableSpace Ω] [MeasurableSpace G] |
| (Q : Kernel Ω G) [IsFiniteKernel Q] (F : G → ℝ) (hF : Measurable F) |
| {B : ℝ} (hB : 0 ≤ B) (h0 : ∀ u, 0 ≤ F u) (hbound : ∀ u, F u ≤ B) |
| (hQ : ∀ ω, Q ω Set.univ ≤ 1) : |
| ∀ ω, 0 ≤ (∫ u, F u ∂Q ω) ∧ (∫ u, F u ∂Q ω) ≤ B := by |
| intro ω |
| have hi : Integrable F (Q ω) := Integrable.of_bound hF.aestronglyMeasurable B |
| (Eventually.of_forall fun u => by simpa only [Real.norm_eq_abs,abs_of_nonneg (h0 u)] using hbound u) |
| refine ⟨integral_nonneg h0,?_⟩ |
| have hh := integral_mono hi (integrable_const B) (fun u => hbound u) |
| rw [integral_const,smul_eq_mul] at hh |
| apply hh.trans |
| have hm : (Q ω).real Set.univ ≤ 1 := by |
| simpa only [measureReal_def,ENNReal.toReal_one] using ENNReal.toReal_mono (by simp : (1:ℝ≥0∞) ≠ ∞) (hQ ω) |
| exact mul_le_of_le_one_left hB hm |
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| lemma integrable_kernel_bounded_integral {Ω G : Type*} [MeasurableSpace Ω] [MeasurableSpace G] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (Q : Kernel Ω G) [IsFiniteKernel Q] |
| (F : G → ℝ) (hF : Measurable F) {B : ℝ} (hB : 0 ≤ B) |
| (h0 : ∀ u, 0 ≤ F u) (hbound : ∀ u, F u ≤ B) |
| (hQ : ∀ ω, Q ω Set.univ ≤ 1) : |
| Integrable (fun ω => ∫ u, F u ∂Q ω) μ := by |
| apply Integrable.of_bound hF.stronglyMeasurable.integral_kernel.aestronglyMeasurable B |
| exact Eventually.of_forall fun ω => by |
| obtain ⟨h0',hB'⟩ := kernel_bounded_integral Q F hF hB h0 hbound hQ ω |
| simpa only [Real.norm_eq_abs,abs_of_nonneg h0'] using hB' |
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| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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