In Riesz Representation Theorem, $X$ is a locally compact Hausdorff space, and $\Lambda$ is a positive linear functional on $C_c(X)$ which is the set of all continuous functions on $X$ with compact support. One conclusion of the theorem is that $\mu(K) < \infty$ for all compact set $K$.
This conclusion is proved in step II (page 43, Rudin's Real and Complex Analysis, third edition). In Rudin's proof, he let $K \prec f$ ($f$ belongs to $C_c(X)$, take value $1$ on $K$, and $0 \leq f \leq 1$), then deduce $\mu(K) \leq \Lambda f$ and assert $\mu(K) < \infty$.
What I don't understand is that can assert $\mu(K) < \infty$ just from $\mu(K) \leq \Lambda f$? Can $\Lambda f$ be infinite?
Please see the exact words below from Rudin.
Since $f\in C_c(X)$ and it is continuous, it is easy to see that $f(X)$ is finite. So, as a result, $\Lambda f$ has to be finite.