This is for anyone who has Rudin's Real and Complex analysis book at hand. I was looking at Rudin's Theorem 2.17 and 2.18. So far everything makes sense, except for one statement that Rudin makes in Theorem 2.18 on page 48.
He states the following: "Since \lambda(K) < \infty for every compact $K$, $\Lambda$ is a positive linear functional on $C_c(X)$,...."
I'm more interested in how he used the hypothesis that \lambda(K) < \infty for all compact set $K$ to conclude that $\Lambda$ is a positive linear function on $C_c(X)$.
At least to my limited knowledge, we are trying to verify that $\Lambda(\alpha f + \beta g) = \alpha \Lambda(f) + \beta \Lambda(g)$ for scalars $\alpha, \beta$ and $f,g \in C_c(X)$. In the first line of Theorem 2.18, Rudin sets $\Lambda f = \int_X f \ d\lambda$ for every $f \in C_c(X)$. Since each $f$ has compact support, then $ \Lambda f = \int_X f \ d\lambda = \int_K f \ d\lambda $. Then from there, I was uncertain how to verify $\Lambda$ is a positive linear functional.
Any tips for this question would be appreciated.