I am studying the proof of Bishop's theorem (generalization of Stone-Weierstrass) in Rudin's Functional Analysis 2nd edition. He make the following statement on the bottom of page 122,
"Since $\mu \to \int g \, d\mu$ is a weak$^*$-continuous function on K, the Krein-Milman theorem implies that $\int g \, d\mu=0$ for every $\mu$ in $K$."
I do not understand why we know that is a weak$^*$-continuous function nor do I see why the Krein-Milman theorem implies that statement. I think one or both these things might have to do with the fact that K is weak$^*$-compact. Thanks for the help!