| 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 BigOperators Topology Classical |
| namespace DirectionalTransience |
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| 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 |
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| end |
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| end |
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| end OAI |
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