Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) What is the dual space of $M(X)$? (2) Suppose $E^*$ is the dual space of $M(X)$, is the following identity true: $$ L(p)=\sup_{f \in E^*} \langle f,p \rangle-L^*(f), $$ where $L^*(f)=\sup_{p\in M(X)}\langle p,f \rangle-L(p)$ is the Legendre transform of $L(p)$.
Although this question is trivial when $M(X)$ is a finite-dimensional space, there seem to be critical gaps to generalize to infinite-dimensional spaces.