Norm of function in $C_0(\Omega)$

Question

Let $(\Omega,\mu)$ be a measure space. For $f\in L^p(\Omega)$, I know that $$\|f\|_p=\sup\left\{\int_\Omega fg\, d\mu:g\in L^q(\Omega), \|g\|_q\leq 1\right\}.$$

I'm wondering whether something similar is true on the space of continuous functions vanishing at infinity:

Let $\Omega$ be locally compact and Hausdorff and $f\in C_0(\Omega)$. Then can we say $$\|f\|_\infty=\sup\left\{\int_\Omega f\, d\mu:\mu\in M(\Omega), \mu(\Omega)\leq 1\right\};$$ where $M(\Omega)$ is the dual of $C_0(\Omega)$?

One side inequality is obvious, but does the other hold as well? A reference would suffice.

Answer 1

In general if you take a real normed space $X$ and its dual $X^*$ then $$||x||_X =\sup_{u\in X^*, ||u||_{X^*} \leq 1} u(x).$$