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180bc03 | 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 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import Mathlib
import OAI.Probability.Ballisticity.Model
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma independent_random_test_weighted_lower {Ω A B : Type*} [MeasurableSpace Ω]
[MeasurableSpace A] [MeasurableSpace B] (μ : Measure Ω) [IsProbabilityMeasure μ]
(U : Ω → A) (V : Ω → B) (hU : Measurable U) (hV : Measurable V)
(hind : IndepFun U V μ) (E : Set A) (hE : MeasurableSet E)
(G : Set (A × B)) (hG : MeasurableSet G) (w : A → ℝ≥0∞) (hw : Measurable w)
(c : ℝ≥0∞) (hbound : ∀ a ∈ E, c ≤ (μ.map V) {b | (a,b) ∈ G}) :
c*(∫⁻ ω in U ⁻¹' E, w (U ω) ∂μ) ≤
∫⁻ ω in {ω | U ω ∈ E ∧ (U ω,V ω) ∈ G}, w (U ω) ∂μ := by
let K : Set (A × B) := (E ×ˢ Set.univ) ∩ G
have hK : MeasurableSet K := (hE.prod MeasurableSet.univ).inter hG
have hUV : Measurable (fun ω => (U ω,V ω)) := hU.prodMk hV
have he : {ω | U ω ∈ E ∧ (U ω,V ω) ∈ G} = (fun ω => (U ω,V ω)) ⁻¹' K := by
ext ω; simp [K]
have hf : Measurable (K.indicator (fun p : A × B => w p.1)) := (hw.comp measurable_fst).indicator hK
have hpush : (∫⁻ ω in {ω | U ω ∈ E ∧ (U ω,V ω) ∈ G}, w (U ω) ∂μ) =
∫⁻ p, K.indicator (fun q => w q.1) p ∂(μ.map U).prod (μ.map V) := by
rw [he,←lintegral_indicator (hK.preimage hUV)]
have heind : (fun ω => ((fun ω => (U ω,V ω)) ⁻¹' K).indicator (fun ω => w (U ω)) ω) =
fun ω => K.indicator (fun p : A × B => w p.1) (U ω,V ω) := by
funext ω
by_cases h : (U ω,V ω) ∈ K <;> simp [h]
rw [heind,←lintegral_map hf hUV,
hind.map_prod_eq_prod_map_map hU.aemeasurable hV.aemeasurable]
rw [hpush,lintegral_prod _ hf.aemeasurable]
have hlow : (∫⁻ a, (E.indicator (fun _ => c*w a)) a ∂μ.map U) ≤
∫⁻ a, ∫⁻ b, K.indicator (fun p => w p.1) (a,b) ∂μ.map V ∂μ.map U := by
apply lintegral_mono
intro a
dsimp only
by_cases ha : a ∈ E
· rw [Set.indicator_of_mem ha]
have heq : (fun b => K.indicator (fun p => w p.1) (a,b)) =
{b | (a,b) ∈ G}.indicator (fun _ => w a) := by
funext b
by_cases hb : (a,b) ∈ G <;> simp [K,ha,hb]
have hslice : MeasurableSet {b | (a,b) ∈ G} := hG.preimage measurable_prodMk_left
rw [heq,lintegral_indicator hslice,lintegral_const]
simp only [Measure.restrict_apply_univ]
calc
c*w a ≤ (μ.map V) {b | (a,b) ∈ G}*w a := by gcongr; exact hbound a ha
_ = w a*(μ.map V) {b | (a,b) ∈ G} := mul_comm _ _
· rw [Set.indicator_of_notMem ha]
exact zero_le
refine LE.le.trans ?_ hlow
rw [←lintegral_indicator (hE.preimage hU)]
have heind2 : (fun ω => (U ⁻¹' E).indicator (fun ω => w (U ω)) ω) =
fun ω => E.indicator w (U ω) := by
funext ω
by_cases h : U ω ∈ E <;> simp [h]
rw [heind2,←lintegral_map (hw.indicator hE) hU]
rw [←lintegral_const_mul c (hw.indicator hE)]
apply le_of_eq
apply lintegral_congr
intro a
by_cases ha : a ∈ E <;> simp [ha]
end DirectionalTransience
end
end
end OAI
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