I'm attending a course on measure theory this semester. While proposing different kinds of convergence (in measure, almost everywhere and in $L^{p}$), our professor stressed (and proved) the fact that convergence almost everywhere is not topological, but claimed that convergence in measure is.
As pointed out in a few questions on this site and wikipedia (1, 2, 3), in the case where $(\Omega, \mathcal{F}, \mu)$ is a finite measure space, convergence in measure can be described by a pseudometric (hence a topology). However, I haven't found an answer to why at least a topology should exist in the case where $\mu$ is an arbitrary measure. Wikipedia (3) claims that their pseudometric works for arbitrary measures, but their proposed function can take $\infty$ as a value, which I believe isn't allowed for metrics.
To sum up: let $(\Omega, \mathcal{F}, \mu)$ be a (not necessarily finite) measure space, does there exist a topology $\mathcal{T}$ on the set of measurable functions $f : \Omega \to \mathbb{R}$ such that a sequence of measurable functions $(f_{n})_{n}$ converges to a measurable function $f$ in measure if and only if it converges to $f$ in the topology $\mathcal{T}$? Extra: Is this topology unique?
Thank you for your help! I've had introductory courses in topology (metric spaces), Banach (Hilbert) spaces and now measure theory.