Corollary 1

For every real measure $\nu$ on $T$, there exist two disjoint sets $M, N$ carrying $\nu^+$ and $\nu^-$, respectively.

One must take care not to confuse the notion of support of a measure $\nu$, and that of a set where $\nu$ is concentrated. The support S of $\nu$ is the smallest closed set carrying $\nu$ (Ch. III, § 2, No. 2, Prop. 2 and Ch. IV, §2, No. 2, Prop. 5). However, there may exist subsets of S that are distinct from S and carry $\nu$. More generally, one can have $\inf(\mu, \nu) = 0$ for two positive measures $\mu$ and $\nu$ having the same support (Exer. 5).

Note also that the intersection of the sets carrying $\nu$ is the set of points $t \in T$ such that $|\nu|(\{t\}) > 0$, and it can be empty (for example, in the case of Lebesgue measure); therefore, there is not in general a smallest set carrying $\nu$.