Perfect measure

In mathematics &mdash; specifically, in measure theory &mdash; a perfect measure (or, more accurately, a perfect measure space) is one that is "well-behaved" in some sense. Intuitively, a perfect measure &mu; is one for which, if we consider the pushforward measure on the real line R, then every measurable set is "&mu;-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 (X, &Sigma;, &mu;) is said to be perfect if, for every &Sigma;-measurable function f : X → R and every A &sube; R with f&minus;1(A) &isin; &Sigma;, there exist Borel subsets A1 and A2 of R such that


 * $$A_{1} \subseteq A \subseteq A_{2} \mbox{ and } \mu \big( f^{-1} ( A_{2} \setminus A_{1} ) \big) = 0.$$

Results concerning perfect measures

 * If X is any metric space and &mu; is an inner regular (or tight) measure on X, then (X, BX, &mu;) is a perfect measure space, where BX denotes the Borel &sigma;-algebra on X.