Signed and complex measures

Relaxation of the requirement of a measure that it be nonnegative yields what is usually called a signed measure. Specifically this is an extended real-valued, countably additive set function μ on a class ε (containing ∅), such that μ(∅) = 0, and such that μ assumes at most one of the values +∞ and −∞ on ε. As for measures, a signed measure μ defined on a class ε, is called finite on ε if |μ(E)| < ∞, for each E ∈ ε, and σ-finite if for each E ∈ ε there is a sequence {En}∞n=1 of sets in ε with E ⊂ ∪∞n=1En and |μ(En)| < ∞, that is, if E can be covered by the union of a sequence of sets with finite (signed) measure. It will usually be assumed that the class on which μ is defined is a σ-ring or σ-field.

Some of the important properties of measures (see Section 2.2) hold also for signed measures. In particular a signed measure is subtractive and continuous from below and above. The basic properties of signed measures are given in the following theorem.

Theorem 5.1.1Let μ be a signed measure on a σ-ring S.