Exercise 6, § 7

Let $X$ be a nonempty compact space, $\mathscr{H}$ a linear subspace of $\mathscr{C}(X; \mathbf{R})$ containing the constants and separating the points of $X$. The relation $\lambda \prec \mu$ defined by $\mathscr{H}$ (Exer. 2) is then an equivalence relation, and the points $\mathscr{H}$-extremal in the sense of Exer. 3 are identical to the points $\mathscr{H}$-extremal in the sense of No. 3. Let $F$ be a closed subset of $X$ containing $\mathrm{Ch}_{\mathscr{H}}(X)$; for every $x \in X$, denote by $\mathscr{M}_x^F$ the set of positive measures $\mu$ on $X$, of total mass 1, such that $\varepsilon_x \prec \mu$ and $\mathrm{Supp}(\mu) \subset F$; in order that $\varepsilon_x \in \mathscr{M}_x^F$, it is necessary and sufficient that $x \in F$; the relation $x \in \mathrm{Ch}_{\mathscr{H}}(X)$ is equivalent to $\mathscr{M}_x^F = \{\varepsilon_x\}$. For every bounded numerical function $f$ defined on $F$ and for every $x \in X$, set

which are finite numbers.

a) Show that the mapping $f \mapsto \overline{H}_x^F(f)$ of $\mathscr{C}(F; \mathbf{R})$ into $\mathbf{R}$ is increasing, positively homogeneous and convex. If $h \in \mathscr{H}$, then $\overline{H}_x^F(h) = h(x)$.

b) For every $x \in X$, every measure $\mu \in \mathscr{M}_x^F$ and every function $f \in \mathscr{C}(F; \mathbf{R})$, $\underline{H}_x^F(f) \leq \int f d\mu \leq \overline{H}_x^F(f)$. Conversely if, for a function $f_0 \in \mathscr{C}(F; \mathbf{R})$, $\gamma$ is a real number such that $\underline{H}_x^F(f_0) \leq \gamma \leq \overline{H}_x^F(f_0)$, then there exists a measure $\mu \in \mathscr{M}_x^F$ such that $\gamma = \int f_0 d\mu$.

c) Show that for a function $g \in \mathscr{C}(X; \mathbf{R})$, the following conditions are equivalent: $\alpha$ For every $x \in X$ and every measure $\mu \in \mathscr{M}_x^F$, one has $g(x) = \int g d\mu$. $\beta$ For every $x \in X$, $\underline{H}_x^F(g|F) = \overline{H}_x^F(g|F) = g(x)$. $\gamma$ For every $\varepsilon > 0$ there exist two finite sequences $(h_i'), (h_j'')$ of functions in $\mathscr{H}$ such that, setting $h' = \sup(h_i')$, $h'' = \inf(h_j'')$, one has $h' \leq g \leq h''$ and $h'' - h' \leq \varepsilon$. (To see that $\beta$ implies $\gamma$), argue as in GT, X, §4, No. 1, Prop. 2.)

When a function $g \in \mathscr{C}(X; \mathbf{R})$ has the preceding equivalent properties, it is said to be $\mathscr{H}$-harmonic; the set $\mathscr{H}^c$ of $\mathscr{H}$-harmonic functions is a closed linear subspace of $\mathscr{C}(X; \mathbf{R})$ containing $\mathscr{H}$; it is independent of the closed set $F \supset \mathrm{Ch}_{\mathscr{H}}(X)$ under consideration. One has $(\mathscr{H}^c)^c = \mathscr{H}^c$, and the equivalence relation in $\mathscr{M}_+(X)$ defined by $\mathscr{H}^c$ is identical to the relation $\lambda \prec \mu$ defined by $\mathscr{H}$; consequently $\mathrm{Ch}_{\mathscr{H}^c}(X) = \mathrm{Ch}_{\mathscr{H}}(X)$. The mapping $g \mapsto g|F$ is a strictly increasing isometry of $\mathscr{H}^c$ onto its image in $\mathscr{C}(F; \mathbf{R})$ (which is therefore a closed subspace of $\mathscr{C}(F; \mathbf{R})$).

d) Take for $X$ the interval $[0,1]$ of $\mathbf{R}$, and for $\mathscr{H}$ the vector space of restrictions to $X$ of the polynomials of second degree over $\mathbf{R}$. Show that $\mathscr{H}^c$ is distinct from the closure $\overline{\mathscr{H}}$ of $\mathscr{H}$ in $\mathscr{C}(X; \mathbf{R})$.

e) Show that in order that $\mathrm{Ch}_{\mathscr{H}}(X) = X$, it is necessary and sufficient that $\mathscr{H}^c = \mathscr{C}(X; \mathbf{R})$ (to prove that the condition is necessary, use $b$).

f) Show that if $\mathscr{H}$ is lattice-ordered (in other words, is a Riesz space), then $\mathscr{H}^c = \overline{\mathscr{H}}$. Give an example where $\mathscr{H}$ is not lattice-ordered and $\mathscr{H}^c = \overline{\mathscr{H}}$.

g) Let $\mathscr{E}_{\mathscr{H}}$ be the smallest closed subset of $\mathscr{C}(X; \mathbf{R})$ containing $\mathscr{H}$ such that the lower envelope $\inf(u, v)$ of any two functions of $\mathscr{E}_{\mathscr{H}}$ belongs to $\mathscr{E}_{\mathscr{H}}$. Show that the following properties are equivalent: $\alpha$ $f \in \mathscr{E}_{\mathscr{H}}$; $\beta$ for every $x \in X$ and every measure $\mu \in \mathscr{M}_x^X$, $\int f d\mu \leq f(x)$; $\gamma$ $\overline{H}_x^X(f) = f(x)$ for every $x \in X$; $\delta$ for every $\varepsilon > 0$, there exists a finite sequence $(h_i)$ of functions in $\mathscr{H}$ such that $f \leq \inf(h_i) \leq f + \varepsilon$. (To see that $\beta$ implies $\gamma$), make use of $b$.) Deduce from this that $\mathscr{E}_{\mathscr{H}}$ is a pointed convex cone, and that $\mathscr{E}_{\mathscr{H}} \cap (-\mathscr{E}_{\mathscr{H}}) = \mathscr{H}^c$. Show that every function in $\mathscr{E}_{\mathscr{H}}$ attains its infimum in $X$ at at least one point of $\mathrm{Ch}_{\mathscr{H}}(X)$.