Perfect measure


In mathematics — specifically, in measure theory — a perfect measure is one that is “well-behaved” in some sense. Intuitively, a perfect measure μ is one for which, if we consider the pushforward measure on the real line R, then every measurable set is “μ-approximately a Borel set”. The notion of perfectness is closely related to tightness of measures: indeed, in metric spaces, tight measures are always perfect.

Definition

A measure space is said to be perfect if, for every Σ-measurable function f : X → R and every A ⊆ R with f−1 ∈ Σ, there exist Borel subsets A1 and A2 of R such that

Results concerning perfect measures