Exercise 31, § 7

Let $X$ be the compact interval $[0, 1]$ of $\mathbf{R}$ and $\Phi$ the set formed from the continuous real valued functions defined in $X$ and the functions $t \mapsto |t - a|^{-\alpha}$, where $a \in X$ and $0 < \alpha < 1$ (we put $0^{-\alpha} = + \infty$ for $\alpha > 0$). In the space $\mathscr{M}(X)$ of measures on $X$, let $\mathscr{M}_\Phi^+$ be the set of the measures $\mu \geq 0$ such that all the functions of $\Phi$ are $\mu$-integrable. a) Give $\mathscr{M}_\Phi^+$ the uniform structure induced by the product structure of $\mathbf{R}^\Phi$. Show that $\mathscr{M}_\Phi^+$ is a proper convex complete cone for this uniform structure. (Note that for every function $f \in \Phi$ there exists $g \in \Phi$ such that, for all $\varepsilon > 0$, there exists $u \in \mathscr{C}(X; \mathbf{R})$ such that $0 \leq f - u \leq \varepsilon g$.) b) Show that the cone $\mathscr{M}_\Phi^+$ has no extremal generator. (Observe that if $\mu \in \mathscr{M}_\Phi^+$, then all the measures $\lambda$ such that $0 \leq \lambda \leq \mu$ belong to $\mathscr{M}_\Phi^+$.) c) Show that the set $S$ of the $\mu \in \mathscr{M}_\Phi^+$ such that $\mu(1) = 1$ is a sole of the cone $\mathscr{M}_\Phi^+$ and a simplex in $\mathbf{R}^\Phi$ (II, p. 71, exerc. 41). \*