I've spent some time searching for this online - both on this site and elsewhere - and even after consulting a considerable amount of literature, I can't seem to nail down an answer. Perhaps someone can help me.
Suppose I'm dealing with the measure space $(\mathbb{R},\Sigma,\mu)$ where $\Sigma$ is the Borel $\sigma$-algebra generated by open sets and where $\mu$ is Lebesgue measure. It's a relatively standard exercise to show that the dual of $L^p$ is $L^q$ where $q=\frac{p}{p-1}$ when $1\leq p<\infty$; one can also show that the dual of $L^\infty$ is (in Royden's notation) $\mathcal{BFA}(\mathbb{R},\mu)$ (i.e.the normed linear space of finitely additive signed measures on $\Sigma$ that are absolutely continuous with respect to $\mu$). Finally, in the case of $0 < p < 1$, the dual of $L^p$ is trivial (i.e., is equal to $\{0\}$); this is shown in any number of papers (e.g., by Day, Conrad, Rosenzweig, etc.). This brings me to my question:
If we write $L^0$ for the space of all measurable functions on the measure space in question, do we know what the dual is? I've read about ba-spaces and the dual of $B(\Sigma)$ (that is, the space of all bounded $\Sigma$-measurable functions with the uniform norm) but that info doesn't work here. Also, a MSE question here discusses something which may be related, but my knowledge of probability-related measure theory is scant.
Any information and/or direction to online resources discussing this would be hugely appreciated.
Edit - Per the comment of @Ian below, the topology I'm considering on $L^0$ is the topology induced by convergence in measure as mentioned in the Wiki.