Exercise 5, § 7

The notations being those of Exer. 4 e), let $\mathscr{A}'$ denote the dual of the normed space $\mathscr{A}$, equipped with the order structure deduced from that of $\mathscr{A}$ (Ch. II, §2). On the other hand, denote by C the convex cone with vertex 0 generated by $X \times \{1\}$ in the space $E \times \mathbf{R}$.

a) Show that the following properties are equivalent: (i) Every point $x \in X$ is the barycenter of a unique maximal measure $\beta_x$ of mass 1 on $X$. (ii) The vector space $F$ generated by $C$ is a lattice for the order structure having $C$ as the set of elements $\geqslant 0$ (in other words, $X$ is a simplex (TVS, II, §2, Exer. 41)). (iii) For every function $f \in \mathscr{P}$, $S(f)$ is both concave and convex. (iv) If $\mu$ and $\nu$ are two maximal measures such that $\mu(h) = \nu(h)$ for all $h \in \mathscr{A}$, then $\mu = \nu$. (v) The vector space $\mathscr{A}'$ is a Riesz space. (vi) If $f$ and $g$ are two functions in $\mathscr{P}$, then $S(f + g) = S(f) + S(g)$. (First prove that (i) implies the following property: (vii) The mapping $x \mapsto \beta_x$ may be extended in only one way to a bijective linear mapping of $F$ onto the subspace of $\mathscr{M}(X)$ generated by the maximal measures, and this mapping transforms $C$ into the cone of maximal measures. Then deduce (ii) from (vii), using Exer. 2 e). To deduce (iii) from (ii), use the decomposition lemma in the Riesz space $F$ and Exer. 4 f). To deduce (iv) from (iii), use the fact that if $g \in \mathscr{C}(X; \mathbf{R})$ is both concave and convex, then the set of $h \in \mathscr{A}$ such that $h(x) > g(x)$ for all $x \in X$ is a decreasing directed set (TVS, II, §5, Prop. 6) and has $g$ for its lower envelope; then apply Exer. 4 e). To deduce (v) from (iv), observe that $\mathscr{A}'$ may be identified with the subspace of $\mathscr{M}(X)$ generated by the maximal measures, with the help of Exer. 4 e). To deduce (vi) from (v), apply the decomposition lemma in the Riesz space $\mathscr{A}'$ and Exer. 4 f). Finally, to deduce (i) from (vi), consider the mapping $f \mapsto S(f)(x)$ for $f \in \mathscr{P}$ and $x \in X$, and use Exer. 4 e).) b) When the conditions of a) are satisfied, show that for every function $f \in \mathscr{C}(X; \mathbf{R})$, the mapping $x \mapsto \beta_x(f)$ is in the closure, for the topology of uniform convergence in $X$, of the set of bounded functions on $X$ that are the difference of two upper semi-continuous functions (make use of the fact that $\mathscr{P}$ is total in $\mathscr{C}(X; \mathbf{R})$); consequently, this mapping is measurable for every measure on $X$. For every positive measure $\mu$ on $X$, the unique maximal measure $\nu \succ \mu$ is given by $\nu = \int \beta_x d\mu(x)$ (cf. Ch. V). c) When the conditions of a) are satisfied and $X$ is in addition metrizable, show that for every $x \in X$, $\beta_x$ is the only positive measure $\mu$ of mass 1 and barycenter $x$ such that $\mu(X - L) = 0$, where $L$ is the set $\mathrm{Ch}_{\mathscr{A}}(X)$ of extremal points (argue as in Th. 1). d) When the conditions of a) are satisfied, show that the following properties are equivalent: $\alpha)$ The set $L = \mathrm{Ch}_{\mathscr{A}}(X)$ is closed, in other words $\check{S}_{\mathscr{A}}(X) = \mathrm{Ch}_{\mathscr{A}}(X)$. $\beta)$ For every function $f \in \mathscr{P}$, the restriction of $S(f)$ to $\overline{L}$ is continuous. $\gamma)$ For every function $f \in \mathscr{P}$, the function $S(f)$ is continuous on $X$. $\delta)$ The mapping $x \mapsto \beta_x$ of $X$ into $\mathscr{M}_+(X)$ (equipped with the vague topology) is continuous. (To show that $\alpha$) implies $\beta)$, observe that $\beta_x = \varepsilon_x$ in $L$. To prove that $\beta)$ implies $\gamma)$, use the fact that $S(f)$ is the lower envelope of a decreasing directed set of functions in $\mathscr{A}$ (TVS, II, §5, Prop. 6), Dini’s theorem and Exer. 3 d) applied to the functions in $\mathscr{A}$. To prove that $\gamma)$ implies $\delta)$, use Exer. 4 e). Finally, to see that $\delta)$ implies $\alpha)$, use Exer. 4 a).)