# Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set

I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to cite. The statement is:

Let $$M$$ be the space of signed Radon measures on compact set $$K\subset\mathbb{R}^n$$, of total variation bounded by 1, endowed with the weak* topology (i.e. $$\mu_n \to \mu$$ in $$M$$ means $$\int gd\mu_n \to \int gd\mu$$ for all $$g\in C(K)$$). Show that $$M$$ is metrizable (with respect to this topology).

Thank you.

• Think about your possible reader! According to your advisor she/he would read According to [17, Lemmas 3.102 and 3.103] $M$ is metrizable. The alternative is Because of Banach-Alaoglu und the separability of $C(K)$, $M$ is metrizable. Jul 17 '19 at 7:26