I am having difficulty understanding a step in the proof of Riesz Representation Theorem, in Rudin's 'Real and Complex Analysis' (P.40, Theorem 2.14):
Let $X$ be a locally compact Hausdorff space, and let $\Lambda$ be a positive linear functional on $C_c(X)$ (the set of all continuous functions on X with compact support)
Rudin defines $$\mu(V)=sup\{\Lambda f:f\prec V\}$$ for every open set $V$ in $X$, where $f\prec V$ means $f$ is continuous and has compact support and $0\le f\le \chi_V$
He goes on to assert that $\mu(E)=inf\{ \mu(V): E\subseteq V, V \:open\}$ for every open set $E$ in $X$
But I have difficulty understanding why the above equality holds. I can prove $\mu(E)\leq inf\{ \mu(V): E\subseteq V, V \:open\}$ immediately from the definition of $\mu$ but the other side of the inequality is giving me trouble.
Any help in elaborating Rudin's statement is very much appreciated.
$E$ is open and $E\subseteq E$, so $\mu(E)\in\{\mu(V):E\subseteq V, V \mathrm{open}\}$. Hence, $\mu(E)\geq \inf\{\mu(V):E\subseteq V, V \mathrm{open}\}$.