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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.

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    $\begingroup$ 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. $\endgroup$
    – Jochen Wengenroth
    Commented Jul 17, 2019 at 7:26

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You can cite Lemma 3.102 and Lemma 3.103 in M. Fabian et al., Banach Space Theory, Springer 2011. Probably there are very many sources that contain this result.

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