Theorems · Theorem · measure theory

MeasureTheory.ProbabilityMeasure.continuous_map

∀ {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'] {f : Ω → Ω'}
  (f_cont : Continuous f), Continuous fun ν => ν.map ⋯

If f : X → Y is continuous and Y is equipped with the Borel sigma algebra, then the push-forward of probability measures f* : ProbabilityMeasure X → ProbabilityMeasure Y is continuous (in the topologies of convergence in distribution).

Defined in

Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Cited by

0 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceTopologicalSpaceOpensMeasurableSpaceTopologicalSpaceBorelSpace