Theorems · Theorem · measure theory

MeasureTheory.ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous

∀ {Ω : Type u_1} {Ω' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace Ω'] [inst_2 : TopologicalSpace Ω]
  [inst_3 : OpensMeasurableSpace Ω] [inst_4 : TopologicalSpace Ω'] [inst_5 : BorelSpace Ω'] {ι : Type u_3}
  {L : Filter ι} (νs : ι → MeasureTheory.ProbabilityMeasure Ω) (ν : MeasureTheory.ProbabilityMeasure Ω),
  Filter.Tendsto νs L (nhds ν) →
    ∀ {f : Ω → Ω'} (f_cont : Continuous f), Filter.Tendsto (fun i => (νs i).map ⋯) L (nhds (ν.map ⋯))

If f : X → Y is continuous and Y is equipped with the Borel sigma algebra, then convergence (in distribution) of ProbabilityMeasures on X implies convergence (in distribution) of the push-forwards of these measures by f.

Defined in

Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Cited by

2 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceTopologicalSpaceOpensMeasurableSpaceTopologicalSpaceBorelSpace