Proposition 13

— Let m be a complex measure, and $\mu = |m|$. For m to be bounded, it is necessary and sufficient that $\mu$ be bounded, in which case $\|m\| = \|\mu\|$.

We have $m = g \cdot \mu$, where $g$ is $\mu$-measurable and $|g(t)| = 1$ for all $t \in T$. If $\mu$ is bounded then, for every function $f \in \mathscr{K}_\mathbf{C}(T)$,

therefore $m$ is bounded and $\|m\| \leq \|\mu\|$. If $m$ is bounded, then for every $f \in \mathscr{K}_\mathbf{C}(T)$ we have, on taking into account Lemma 5, therefore $\mu$ is bounded and $\|\mu\| \leq \|m\|$. Whence the proposition.