Say that $(X,M)$ is a measurable space (i.e., $0,X \in M$ and $A \in M \implies X \setminus A \in M$ and, finally, $M$ is closed w.r.t. countable unions/intersections). Suppose that $\mu$ is either a signed or complex measure on $(X,M)$. In particular, $\mu$ is a function on $M$.
I know that if $\mu$ maps into $[0,\infty)$, then $d(A,B) = \mu(A \Delta B)$ is a metric modulo the equivalence relation $A \sim B \iff \mu(A\Delta B) = 0$. This was mentioned in baby Rudin, for instance.
More generally, however, if we are given a topology $T$ on $X$ does there consequently exist a canonical topology on $M$ induced by $T$ such that $\mu$ is continuous?
Of course, I would prefer if the topology on $M$ were not just the discrete topology.
Further, it would be nice if it were metrizable. If the answer is affirmative, a follow up question might be, for instance, "how coarse can the topology be while still making $\mu$ continuous?" Has anyone thought about this or seen this before?