Corollary 16.1.7. Let $(X, \cm)$ be a measurable space, then
For any signed measure $\mu: \cm \to (-\infty, \infty)$, $\mu$ is bounded.
For any complex measure $\mu: \cm \to \complex$, $\mu$ is bounded.
Proof. (1): Let $X = P \sqcup N$ be a Hahn decomposition of $\mu$, then for any $E \in \cm$, $\mu(E) \in [\mu(N), \mu(P)]$.
(2): Since $\text{Re}(\mu)$ and $\text{Im}(\mu)$ are both signed measures taking values in $(-\infty, \infty)$, they are bounded by (1). Thus $\mu$ is also bounded.$\square$