| import OAI.Probability.InvariantIsing.Spectral.CompactResolventTests | |
| namespace OAI | |
| /-! A direct coupling bound for the Levy--Prokhorov distance of two images. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set Metric | |
| namespace InvariantIsing | |
| theorem levyProkhorov_map_dist_le_ae {Ω E : Type*} [MeasurableSpace Ω] [PseudoMetricSpace E] | |
| [MeasurableSpace E] [BorelSpace E] (P : Measure Ω) [IsProbabilityMeasure P] | |
| (f g : Ω → E) (hf : Measurable f) (hg : Measurable g) | |
| {δ : ℝ} (hδ : 0 ≤ δ) (hfg : ∀ᵐ ω ∂P, dist (f ω) (g ω) ≤ δ) : | |
| levyProkhorovDist (P.map f) (P.map g) ≤ δ := by | |
| have : IsProbabilityMeasure (P.map f) := (Measure.isProbabilityMeasure_map_iff hf.aemeasurable).mpr inferInstance | |
| have : IsProbabilityMeasure (P.map g) := (Measure.isProbabilityMeasure_map_iff hg.aemeasurable).mpr inferInstance | |
| apply levyProkhorovDist_le_of_forall_le _ _ hδ | |
| intro ε S hε hS | |
| rw [Measure.map_apply hf hS,Measure.map_apply hg isOpen_thickening.measurableSet] | |
| have hs : f ⁻¹' S ≤ᵐ[P] g ⁻¹' thickening ε S := by | |
| filter_upwards [hfg] with ω hω hS' | |
| apply mem_thickening_iff.mpr | |
| exact ⟨f ω,hS',(by rw [dist_comm]; exact hω.trans_lt hε)⟩ | |
| exact (measure_mono_ae hs).trans (le_add_right le_rfl) | |
| end InvariantIsing | |
| end | |
| end OAI | |