Definition 20.1.1 (Signed Measure).label Let $(X, \cm)$ be a measurable space and $\mu: \cm \to [-\infty, \infty]$, then $\mu$ is a signed measure if
- (M1)
$\mu(\emptyset) = 0$.
- (M2)
For any $\seq{E_n}\subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}E_n}= \sum_{n \in \natp}\mu(E_{n})$ where the sum converges absolutely.
By Riemann’s Rearrangement Theorem, (M2) implies that $\mu$ can only take at most one value in $\bracs{-\infty, \infty}$.