In mathematics — specifically, in measure theory — a **perfect measure** (or, more accurately, a **perfect measure space**) 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.

A measure space (*X*, Σ, *μ*) is said to be **perfect** if, for every Σ-measurable function *f* : *X* → **R** and every *A* ⊆ **R** with *f*^{−1}(*A*) ∈ Σ, there exist Borel subsets *A*_{1} and *A*_{2} of **R** such that

- If
*X*is any metric space and*μ*is an inner regular (or tight) measure on*X*, then (*X*,*B*_{X},*μ*) is a perfect measure space, where*B*_{X}denotes the Borel*σ*-algebra on*X*.