Proposition 11

— If $\mu$ and $\nu$ are two positive measures on $X$, then $\| \mu + \nu \| = \| \mu \| + \| \nu \|$.

For, the functions $f \in \mathscr{K}(X; \mathbf{R})$ such that $0 \leq f \leq 1$ form a directed set $S$ for the relation $\leq$. For a positive measure $\mu$ on $X$, it therefore follows from (22) and the monotone limit theorem that $\| \mu \| = \lim_{f \in S} \mu(f)$; the conclusion of the proposition then follows at once.