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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 A1 and A2 of R such that
Results concerning perfect measures
- If X is any metric space and μ is an inner regular (or tight) measure on X, then (X, BX, μ) is a perfect measure space, where BX denotes the Borel σ-algebra on X.