Proposition 8

If $\mu$ is a measure on $X$ then, for every function $h \in \mathscr{C}(X; \mathbf{C})$,

For, if $f \in \mathscr{K}_+(X)$ and if $g \in \mathscr{K}(X; \mathbf{C})$ is such that $|g| \leq f$, then, by (13), $|\int gh d\mu| \leq \int |gh| d|\mu| \leq \int f|h| d|\mu|$, which proves (18).