$\nu(A)=\int \mathcal{X}_{A}f(x)d\mu$ measure associated functional $l_{\nu}(g)=\int g(x)f(x)d\mu.$

Question

Let $\mu$ be a positive Baire measure, $f\in L^1(X,d\mu)$ with $X$ compact hausdorff and $f\geq 0$. For any Baire set $A$, define $\nu(A)=\int \mathcal{X}_{A}f(x)d\mu$. Show that $\nu$ is the one measure associated to the functional on $C(X)$,

$l_{\nu}(g)=\int g(x)f(x)d\mu.$

Hello! In general, what are the steps to demonstrate that a measure comes from a functional one?