Proposition 15

Every complex measure $\sigma$ on $T$ may be written in one and only one way in the form $\rho + \theta$, where $\rho$ is an atomic measure and $\theta$ is a diffuse measure; one then has $|\sigma| = |\rho| + |\theta|$..

The uniqueness of the decomposition is obvious since, $\rho$ being atomic and $\theta$ diffuse, necessarily $\rho = \sum_{t \in T} \rho(\{t\}) \varepsilon_t = \sum_{t \in T} \sigma(\{t\}) \varepsilon_t$ and $\theta = \sigma - \rho$. To establish existence, it suffices to observe that $\sum_{t \in K} |\sigma(\{t\})| \leq |\sigma|(K) < +\infty$ for every compact set $K$, so that one can set $\sum_{t \in T} \sigma(\{t\}) \varepsilon_t = \rho$. The measure $\sigma - \rho$ is obviously diffuse, and the relation $|\sigma| = |\rho| + |\sigma - \rho|$ follows at once from Cor. 3 of Prop. 13 of No. 7.

One observes that this proof shows that if $\sigma$ is carried by a set $M$ and if $|\sigma|(\{t\}) > 0$ for every $t \in M$, then $\sigma$ is atomic.