Let $X$ be a Hausdorff topological space and $C_0(X)$ be the $\Bbb C$ $*$-algebra of continuous functions that vanish at infinity. If $\omega \in C_b (X) ^*$, then there exists a complex, finite, regular Borel measure $\mu$ such that $\omega (f) = \int _X f \ \Bbb d \mu$.
Let's say that I want use $\omega$ to check whether a certain $E \subseteq X$ is a null set of $\mu$. Unfortunately, I cannot evaluate $\omega (1_E)$ because $1_E$ is not continuous. How should I proceed then?