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 "A"₁ and "A"₂ 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", "B"_"X", "μ") is a perfect measure space, where "B"_"X" denotes the Borel "σ"-algebra on "X".

References

* cite book
last = Parthasarathy
first = K. R.
title = Probability measures on metric spaces
publisher = AMS Chelsea Publishing, Providence, RI
year = 2005
pages = pp.xii+276
isbn = 0-8218-3889-X MathSciNet|id=2169627 (See chapter 2, section 4.)