Exercise 2, § 2

Let $(\lambda_n)$ be a sequence of complex measures on T. For the sequence $(\lambda_n)$ to be summable in $\mathscr{M}(T; \mathbf{C})$ equipped with the vague topology, it is necessary and sufficient that, for every function $f \in \mathscr{H}(T)$, the series with general term $\lambda_n(f)$ be absolutely convergent (make use of the Banach-Steinhaus theorem; cf. TVS, III, §4, No. 2, Cor. 2 of Th. 1).