Representation of Functional in the Dual of the Space of Signed Measures on a Measurable Space

Question

I suspect the following is true, but I don't know a reference. Let $X$ be a measurable space (with $\sigma$-algebra $\mathcal{F}(X)$), and let $MX$ denote the space of finite signed measures on $X$. Then \begin{equation*} \phi(\nu)=\int_X\phi(\delta_x)\,d\nu(x), \end{equation*} for all $\phi\in(MX)'$ and $\nu\in MX$, where $\delta_x\in MX$ is the Dirac measure concentrated at $x\in X$.

Answer 1

No. Let $X=[0,1]$ with the Borel sets. If $\mu$ is atomless, let $\lambda(\mu)=0$. If $\mu=\sum_{i=1}^\infty\alpha_i\delta_{x_i}$ is discrete, let $\lambda(\mu)=\sum_{i=1}^\infty\alpha_i$. Extend $\lambda$ by linearity to all finite signed measures.

For $\mu$ the uniform distribution, we have $$\lambda(\mu)=0\neq 1=\int 1 \mathrm d\mu=\int \lambda(\delta_x)~\mathrm d\mu(x).$$