I have recently read a theorem in a paper (porbably) indicating that if we take a Banach Lattice $E \subseteq L^1$ with order continuous norm, then we can identify its dual $E'$ with another space $F \subseteq L^1$ by the identity
$$\ell_g (f) = \int fg d\mu, \text{ } \text{ }\text{ } \ell \in E', \text{ } g \in F$$
where $\mu$ is a finite (probability) measure. I know, that this is true for the $L^p$-spaces ($E = L^p$, $F=L^q$) and, more generalized, for Orlicz spaces with finite young function, but I can't reason why this should hold for any other Banach Lattice in $L^1$ with order continuous norm.
Is there a "basic" theorem I do not know? Can someone give me a hint or literature on this? Or does such a theorem not exist? I would be really thankful for an answer, since I am stuck on this for a longer time!