Let $(X, \mu)$ be a space with measure. It is known that a Banach predual of the space of finitely additive measures that are absolutely continuous with respect to $\mu$ is $L^\infty (X, \mu)$.
If I strengthen the additivity requirement, is a Banach predual of the space of countably additive measures (i.e. measures in the usual sense) that are absolutely continuous with respect to $\mu$ known?
You're asking for a predual of $L^1(X,\mu)$, which by Radon-Nikodym corresponds to the absolutely continuous countably additive measures (in the $\sigma$-finite case).
In the case of counting measure on $\mathbb N$, $L^1(X,\mu)$ is the sequence space $\ell^1$, and a predual of this is $c_0$.
In the case of Lebesgue measure on $[0,1]$, there is no predual: see this question and its answer
