Proposition 8

Suppose X is metrizable. Then the set $\operatorname{Ch}_{\mathscr{H}}(X)$ of $\mathscr{H}$-extremal points of X is the intersection of a countable family of open sets in X, and for every $x \in X$, there exists a positive measure $\mu$ of total mass 1 on X such that

for every $h \in \mathscr{H}$.

This is the translation of Th. 1 of No. 2, by transport of structure by means of the homeomorphism $x \mapsto i_{\mathscr{H}}(x)$, as in Prop. 5.

A certain number of results of this No. may be extended when $\mathscr{H}$ is replaced by a set $\mathscr{P}$ of functions defined on X, with values in $\mathbf{R} \cup \{+\infty\}$, that are lower semi-continuous, $\mathscr{P}$ being assumed to contain the constants and to satisfy $\mathscr{P} + \mathscr{P} \subset \mathscr{P}$ (Exer. 2).