Proposition 4

— Let $f$ be a function in $J_+$. The mapping $\mu \mapsto \mu^*(f)$ of the set $\mathscr{M}_+(X)$ of positive measures on $X$, into the extended real line $\overline{\mathbf{R}}$, is lower semi-continuous for the vague topology on $\mathscr{M}_+(X)$ (Ch. III, §1, No. 9).

For, this mapping is by definition the upper envelope of the mappings $\mu \mapsto \mu(g)$, where $g$ runs over the set of functions in $\mathscr{K}_+$ such that $g \leq f$; and by definition of the vague topology, the mappings $\mu \mapsto \mu(g)$ are continuous on $\mathscr{M}(X)$.