Exercise 3, § 7

Let X be a nonempty compact space, $\mathscr{P}$ a set of lower semi-continuous numerical functions on X, taking values in $]-\infty, +\infty]$; assume that $\mathscr{P}$ contains the finite constants. A point $x \in X$ is said to be $\mathscr{P}$-*extremal* if the measure $\varepsilon_x$ is minimal for the pre-order relation $\mu \prec \nu$ [^1]; the set of $\mathscr{P}$-extremal points is denoted $\mathrm{Ch}_{\mathscr{P}}(X)$.

a) Show that if $\mathscr{P}'$ is the set of linear combinations $\sum_i c_i h_i$ of functions in $\mathscr{P}$ with coefficients $\geqslant 0$, the $\mathscr{P}$-extremal points are identical to the $\mathscr{P}'$-extremal points.

b) For a point $x \in X$, show that the following conditions are equivalent: α) $x$ is $\mathscr{P}$-extremal. β) For every function $f \in \mathscr{C}(X; \mathbf{R})$ and every $\varepsilon > 0$, there exists an $h \in \mathscr{P}'$ such that $f \leqslant h$ and $h(x) \leqslant f(x) + \varepsilon$. γ) For every open neighborhood U of $x$ and every $\varepsilon > 0$, there exists a function $h \geqslant 0$ in $\mathscr{P}'$ such that $h(x) \leqslant \varepsilon$ and $h(y) \geqslant 1$ for all $y \in X - U$. (Use Exer. 2 b).)

Show, moreover, that the set of points of $\mathrm{Ch}_{\mathscr{P}}(X)$ where at least one function in $\mathscr{P}$ attains its infimum in X is dense in $\mathrm{Ch}_{\mathscr{P}}(X)$.

c) A subset F of X is said to be $\mathscr{P}$-*stable* if it is closed and if the relations $\lambda \prec \mu$ and $\mathrm{Supp}(\mu) \subset F$ imply $\mathrm{Supp}(\lambda) \subset F$. Show that if $u \in \mathscr{P}$ and if $F'$ is the set of points of F where $u$ attains its infimum in F, then the set $F'$ is $\mathscr{P}$-stable.

d) Assume in addition that $\mathscr{P}$ separates the points of X. Show that for every function $h \in \mathscr{P}$, the set $S_h$ of points of X where $h$ attains its infimum in X intersects $\mathrm{Ch}_{\mathscr{P}}(X)$. (Consider a minimal $\mathscr{P}$-stable subset contained in $S_h$ and use c) to show that such a subset reduces to a single point.)

e) With the same hypotheses as in d), show that for a closed subset F of X to contain $\mathrm{Ch}_{\mathscr{P}}(X)$, it is necessary and sufficient that for every function $h \in \mathscr{P}'$, F intersect the set $S_h$ of points where $h$ attains its infimum in X (make use of b)). From this, deduce that for $a \in X$ to be in the closure of $\mathrm{Ch}_{\mathscr{P}}(X)$, it is necessary and sufficient that for every open neighborhood U of $a$, there exist $h \in \mathscr{P}'$ and $b \in U$ such that $h(b) < h(x)$ for all $x \in X - U$ (in other words, $S_h \subset U$).

f) Under the hypotheses of d), a subset A of X is said to be a $\mathscr{P}$-edge if, for every function $h \in \mathscr{P}'$, $S_h$ intersects A. Take for X the union, in $\mathbf{R}^2$, of the two circles with respective centers $(-1,0)$ and $(1,0)$ and with radius 1, and for $\mathscr{P}$ the set of restrictions to X of the affine linear functions on $\mathbf{R}^2$. Show that $\mathrm{Ch}_{\mathscr{P}}(X)$ consists of the points $(\xi,\eta)$ of X such that $|\xi| \geq 1$. If $a$ is the $\mathscr{P}$-extremal point $(1,1)$, show that there is a neighborhood U of $a$ such that there is no function $h \in \mathscr{P}'$ for which $h \geq 0$, $h(a) = 0$ and $h(y) \geq 1$ on $X - U$. If A and $A'$ are the complements in $\mathrm{Ch}_{\mathscr{P}}(X)$ of the points $(1,1)$ and $(-1,1)$, respectively, then A and $A'$ are $\mathscr{P}$-edges but the same is not true of $A \cap A'$; thus there is no smallest $\mathscr{P}$-edge.

g) Let Y be a second compact space, $\mathscr{Q}$ a set of lower semi-continuous mappings of Y into ]$-\infty, +\infty$], containing the finite constants. Let $\mathscr{H}$ be the set of functions $h = f \otimes g$ with $f \in \mathscr{P}$ and $g \in \mathscr{Q}$; show that $\mathrm{Ch}_{\mathscr{H}}(X \times Y) = \mathrm{Ch}_{\mathscr{P}}(X) \times \mathrm{Ch}_{\mathscr{Q}}(Y)$.

[^1]: The definition of maximal (or minimal) element in an ordered set (S, III, §1, No. 6) extends at once to pre-ordered sets.