Definition of sigma finite signed measure

Question

By the definition I work with, a signed measure is an extended real valued measure, so not necessarily finite. How does one define sigma finiteness? So that both, negative and positive part (by Jordan decomposition) are sigma finite measures?

Answer 1

We can extend the definition of $\sigma$-finite measures naturally to signed measures:

Given a [signed] measure $\mu$ on a space $X,$ we should say $\mu$ is $\sigma$-finite if and only if $X$ can be covered with at most countably many measurable sets of finite measure. Symbolically, this means $X = \bigcup_{n\in \mathbb{N}} A_n$ where each $A_n$ is a measurable set and $[-\infty < ]\ \mu(A_n) < \infty.$

In the above, the parts I've enclosed in brackets [ ] are the only parts that are added to extend the definition.

In addition, with respect to the Jordan decomposition $\mu = \mu^+ - \mu^-,$ the following are equivalent:

1. The signed measure $\mu$ is $\sigma$-finite
2. Both measures $\mu^-, \mu^+$ are $\sigma$-finite
3. The measure $|\mu| = \mu^- + \mu^+$ is $\sigma$-finite.

In light of this, you could make (2.) your definition of a $\sigma$-finite signed measure, if you want.