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.

Notation 1.1 (MEAS ; MEASfinite). 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 MEASfinite.