Proposition 12

Let $(\mu_n)$ be a sequence of bounded positive measures on $T$; there exists a bounded positive measure $\mu$ on $T$ such that the relation $\mu^*(N) = 0$ is equivalent to «$\mu_n^*(N) = 0$ for every $n$»; each of the measures $\mu_n$ has base $\mu$. Moreover, if $\mu'$ is a second positive measure on $T$ having this property, then $\mu$ and $\mu'$ are equivalent.

The last part of the statement follows at once from Def. 3. To prove the existence of $\mu$, we can restrict ourselves to the case that $\mu_n \neq 0$ for every $n$; the family of measures $\mu_n / 2^n \| \mu_n \|$ is then summable in $\mathscr{M}(T)$, and its sum $\mu$ is such that $\| \mu \| \leq 1$. Moreover, since $\mu_n \leq 2^n \| \mu_n \| \cdot \mu$, the relation $\mu(N) = 0$ implies that $\mu_n(N) = 0$ for all $n$; conversely, if $N$ is a set that is negligible for all the $\mu_n$, then it is locally negligible for $\mu$

(§2, No. 2, Cor. 2 of Prop. 1), hence is $\mu$-negligible since $\mu$ is bounded (§1, No. 2, Cor. 2 of Prop. 7).