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c4cdbbd | 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 | 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
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