Example 3

Let F be any set, $\mathscr{U}$ an ultrafilter on F (GT, I, §6, No. 4), E the Riesz space $\mathscr{B}(F)$ of bounded real-valued functions on F. For every $x \in E$, $\lim_{\mathscr{U}} x(t)$ exists, because $x(\mathscr{U})$ is a base of an ultrafilter on the relatively compact set $x(F)$, hence is convergent. Moreover, if $x \geq 0$ then $\lim_{\mathscr{U}} x(t) \geq 0$ by the principle of extension of inequalities; the mapping $x \mapsto \lim_{\mathscr{U}} x$ is thus a positive linear form on E. If $\mathscr{U}$ is taken to be the ultrafilter formed by the sets containing an element $a \in F$, one recovers the positive linear form $x \mapsto x(a)$ (Example 1).