Perfect measure

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.

Definition

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

[math]\displaystyle{ A_{1} \subseteq A \subseteq A_{2} \mbox{ and } \mu \big( f^{-1} ( A_{2} \setminus A_{1} ) \big) = 0. }[/math]

Results concerning perfect measures

• 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.

References

• Parthasarathy, K. R. (2005). "Chapter 2, Section 4". Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. ISBN 0-8218-3889-X. 
• Rodine, R. H. (1966). "Perfect probability measures and regular conditional probabilities". Ann. Math. Statist. 37: 1273–1278. 
• Hazewinkel, Michiel, ed. (2001), "Perfect measure", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=P/p072070