Corollary

— Let $(g_\alpha)$ be a sequence of locally $\mu$-integrable numerical functions, such that the sequence of measures $g_n \cdot \mu$ is increasing. In order that this sequence have an upper bound in the ordered vector space $\mathscr{M}(T)$ of real measures on T, it is necessary and sufficient that the function $g = \sup g_n$ be locally $\mu$-integrable; the supremum in $\mathscr{M}(T)$ of the sequence $(g_n \cdot \mu)$ is then the measure $g \cdot \mu$.

It suffices to apply Prop. 6 to the functions (positive locally almost everywhere) $g'_n = g_{n+1} - g_n$.