In big Rudin's book, the proof of Riesz's representation theorem quite confuses me. It states that the definition of $\mu_1(V)=\mbox{sup}\{Lf:f\prec V\}$ and $\mu_2(V)=\mbox{inf}\{\mu_1(U): V\subset U, U \ \mbox{is open}\}$ is equivalent for all $V\subset X$, where $X$ is a topological space
It is obvious that when $V$ is open. For general $V$, it is not difficult to prove $\mu_2(V)\leq\mu_1(V)$, but I don't know how to prove the opposite direction.