Exercise 4, § 7

Let E be a Hausdorff locally convex space, X a compact convex set in E, $\mathscr{P}$ the set of finite numerical functions continuous and convex in X; $\mu \prec \nu$ denotes the pre-order relation defined on $\mathscr{M}_+(X)$ by $\mathscr{P}$ (Exer. 2).

a) Show that $\mu \prec \nu$ is an order relation on $\mathscr{M}_+(X)$ (use Exer. 29 of TVS, II, §5) and that, for every $x \in X$, the relation $\varepsilon_x \prec \mu$ is equivalent to saying that $\mu$ has mass 1 and $x$ is the barycenter of $\mu$. For every maximal measure $\mu$ (for the preceding order relation) of mass 1, there therefore exists one and only one $x \in X$ such that $\varepsilon_x \prec \mu$. Conversely, every $z \in X$ is the barycenter of at least one maximal measure. For $x \in X$ to be an extremal point of X, it is necessary and sufficient that $\varepsilon_x$ be a maximal measure.

b) Let $\mu$ be a maximal measure on X, $f$ a function in $\mathscr{C}_+(X)$. Show that for every $\varepsilon > 0$, there exists a continuous convex function $g$ on X such that $0 \leq g \leq f$ and $\mu(g) \geq \mu(f) - \varepsilon$ (use Exers. 3 b) and 2 d)). From this, deduce that the support of $\mu$ is contained in the closure of the set of extremal points of X.

c) Let $\mu$ be a maximal measure on X, $(f_n)_{n \geq 1}$ a decreasing sequence of functions in $\mathscr{C}_+(X)$. Show that if, for every extremal point $x \in X$, the sequence $(f_n(x))$ tends to 0, then $\lim_{n \to \infty} \mu(f_n) = 0$ (use b) to construct a decreasing sequence $(g_n)$ of continuous convex functions such that $0 \leq g_n \leq f_n$ and $\mu(g_n) \geq \mu(f_n) - \varepsilon$ for all $n$, and show that the sequence $(g_n(y))$ tends to 0 for every $y \in X$.

d) Deduce from c) that if $A \subset X$ contains no extremal point and is the union of a sequence $(K_n)$ of compact subsets of X, each of which is a countable intersection of open sets in X, then $\mu(A) = 0$ for every maximal measure $\mu$.

e) Let $\mathscr{A}$ be the vector space of continuous functions on X that are the restrictions to X of continuous affine linear functions on E. The relation $\lambda \prec \mu$ implies that $\lambda(h) = \mu(h)$ for every function $h \in \mathscr{A}$; for every function $f \in \mathscr{C}(X; \mathbf{R})$, the lower envelope $S(f)$ of the functions $h \in \mathscr{A}$ such that $f \leq h$ is also the lower envelope of the functions $g \in -\mathscr{P}$ such that $f \leq g$. For every $x \in X$, the relation $\varepsilon_x \prec \mu$ is equivalent to the relation $h(x) = \langle h, \mu \rangle$ for all $h \in \mathscr{A}$; the extremal points of X are identical to the $\mathscr{A}$-extremal points. If $f \in \mathscr{P}$ then, for every $x \in X$,

(*)

(use Exer. 2 b)). This formula is no longer necessarily valid if it is only assumed that $f$ is convex and upper semi-continuous on X (with the notations of Exer. 1 b), consider the function $f$ equal to 1 on the image of I in K, and to 0 elsewhere.

f) Show that the formula (*) is again valid when in the second member one limits oneself to the discrete measures $\mu$ having $x$ as barycenter.