Let $(\Omega,\mathcal{F})$ be a measurable space, $\varphi,\psi_{1},\ldots,\psi_{p}$ be real valued measurable functions on $(\Omega,\mathcal{F})$ and the set $\mathcal{M}$ of all non-negative measures on the measurable space $(\Omega,\mathcal{F})$ such that each function $\varphi,\psi_{1},\ldots,\psi_{p}$ is $\mu$-integrable for all $\mu\in \mathcal{M}$.
Let $\mathcal{X}$ be be the linear space (over $\mathbb{R}$) of signed measures generated by $\mathcal{M}$.
The question: What is the dual space of $\mathcal{X}$? Actually I need a characterization of the dual space.
Remark: In this link it is implied that the dual space of $\mathcal{X}$ is $$\mathcal{X}'=\left\{f:\Omega\rightarrow\mathbb{R}\:|\: f \mbox{ is linear combinations of the functions }\varphi,\psi_{1},\ldots,\psi_{p}\right\}.$$ My intention is to show this last through an isometry.
My attempt: We called $$\mathcal{R}:=\left\{f:\Omega\rightarrow\mathbb{R}\:|\: f \mbox{ is linear combinations of the functions }\varphi,\psi_{1},\ldots,\psi_{p}\right\}.$$
Then we consider the following function $$ \begin{array}{rclrcl} \Phi:\mathcal{R} &\longrightarrow & \mathcal{X}' \\ f=\sum_{i=1}^{p}\alpha_{i}f_{i} & \longmapsto & \Phi f:&\mathcal{X} &\longrightarrow &\mathbb{R} \\ & & &\mu=\sum_{j=1}^{k}\beta_{j}\mu_{j} &\longmapsto & \int_{\Omega} f(\xi) \mu(d\xi). \end{array} $$ I showed that $\Phi$ is well defined, it is linear and injective, the problem is that I have failed to demonstrate the surjectivity, this situation has led me to doubt whether this actually makes some sense.