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9b6e014 | 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 | import Mathlib
import OAI.Probability.Ballisticity.Model
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Topology Classical
namespace DirectionalTransience
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
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'
end DirectionalTransience
end
end
end OAI
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