Perfect measure

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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.

[edit] 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

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

[edit] Results concerning perfect measures

[edit] References

  • Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI, pp.xii+276. ISBN 0-8218-3889-X.  MR2169627 (See chapter 2, section 4.)