How to prove that the dual of $L^\infty(X,\mu)$ is the space of finitely additive, signed measures on $X$ that are absolutely continuous with respect to $\mu$?
I only find that it's a "well known result", both in articles and on wikipedia, but I can't find a single proof of it.
Usually we prove isomorphism with a linear bijective function that preserves the norm, for example if you wanna prove that $(l^1)'=l^\infty$ you find a bijective linear operator $T:(l^1)' \rightarrow l^\infty$ such that $||T(f)||=||f||,\,f\in (l^1)'$. But in this case we're talking about measures, how do we attack it?