Remark 2

Let $(\theta_\alpha)_{\alpha \in A}$ be a family of complex measures on X; the family $(\theta_\alpha)$ is said to be summable if the family $(|\theta_\alpha|)$ of positive measures is summable; it is not sufficient for this that the family $(\theta_\alpha)$ be summable in the vector space $\mathscr{M}(X; \mathbf{C})$ equipped with the vague topology (cf. Exer. 3).