Is there a nice characterization of the predual of $L^1$? So, what does the space $X$ look like, such that $X^*=L^1$, where the star denotes the dual of a Banach space. How do you start to find such preduals in general?
For some context, it is well known that given a measure space $(S, \Sigma, \mu)$, $L^p := L^p(S, \mu)$ is a Banach space for $p\in (1,\infty)$ and that $L^p \cong (L^q)^*$ where $q$ is the Holder conjugate of $p$, that is $\frac 1p + \frac 1q =1$. It is also known that $L^1$ is the predual of $L^\infty$. This leaves the above questions as the only remaining case.
When $S$ is (for example) finite of course the question is moot. If you like one can consider only very simple measure space, like $[0,1]$ with the Lebesgue measure.