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

If X is any metric space and μ is an inner regular measure on X, then is a perfect measure space, where B_X denotes the Borel σ-algebra on X.