The absolute property

Abstract

A measure space M( X, μ ) is a triple ( X, μ, (X, μ) , where μ is a countably additive, nonnegative, extended real–valued function whose domain is the σ–algebra (X, μ) of subsets of a set X and satisfies the usual requirements. A subset M of X is said to be μ–measurable if M is a member of the μ–algebra M(X, μ). For a separable metrizable space X , denote the collection of all Borel sets of X by B(X) . A measure space M(X, μ) is said to be Borel if B(X) ⊂ M(X, μ) , and if M ∈ M(X, μ) then there is a Borel set B of X such that M ⊂ B and μ(B) = μ(M) 1 . Note that if μ(M) , then there are Borel sets A and B of X such that A ⊂ M ⊂ B and μ(B A) = 0. Certain collections of measure spaces will be referred to often – for convenience, two of them will be defined now. N otation 1.1 (MEAS ; MEAS finite ). The collection of all complete, σ–finite Borel measure spaces M(X, μ) on all separable metrizable spaces X will be denoted by MEAS. The subcollection of MEAS consisting of all such measures that are finite will be denoted by MEAS finite .