Exercise 8, § 7

The general notations and hypotheses being those of Exer. 6, set $S = \check{S}_{\mathscr{H}}(X)$. A function $f \in \mathscr{C}(S; \mathbf{R})$ is said to be $\mathscr{H}\text{-resolvent}$ if there exists a function $g \in \mathscr{H}^c$ such that $f = g|S$ (this function is then unique).

a) Show that the following conditions on a function $f \in \mathscr{C}(S; \mathbf{R})$ are equivalent: (i) $f$ is $\mathscr{H}\text{-resolvent}$. (ii) For every $x \in X$, $\underline{H}_x^S(f) = \overline{H}_x^S(f)$. (iii) For every $x \in X$ and every pair of measures $\mu_1, \mu_2$ in $\mathscr{M}_x^S$, one has $\int f d\mu_1 = \int f d\mu_2$. (Make use of Exer. 6 b) and c).) b) Show that the following properties are equivalent: (i) Every function $f \in \mathscr{C}(S; \mathbf{R})$ is $\mathscr{H}\text{-resolvent}$. (ii) For every $x \in X$ and every function $f \in \mathscr{C}(S; \mathbf{R})$, $\underline{H}_x^S(f) = \overline{H}_x^S(f)$. (iii) The set $\mathscr{M}_x^S$ reduces to a single element. (iv) The set $\mathscr{H}^c$ is lattice-ordered. (v) For every function $u \in \mathscr{E}_{\mathscr{H}}$ (Exer. 6 g)), there exists a greatest lower bound $h_u$ of $u$ in $\mathscr{H}^c$. (To show that (iv) implies (i), apply Exer. 7 d) to $\mathscr{H}^c$, proving that for every point $s \in S$, the linear form $h \mapsto h(s)$ is latticial in $\mathscr{H}^c$; then use Stone’s theorem and Exer. 6 c). To see that (i) implies (v), note that $\mathscr{H}^c \subset \mathscr{E}_{\mathscr{H}}$ and consider the unique function $h_u \in \mathscr{H}^c$ such that $h_u|S = u|S$. To see that (v) implies (iv), note that if $h \in \mathscr{H}^c$ then $-h^- = \inf(h, 0) \in \mathscr{E}_{\mathscr{H}}$, and apply (v) to $u = -h^-$.)

c) If the conditions of b) are satisfied and if $\gamma_x$ is the unique element of $\mathscr{M}_x^S$, show that the mapping $x \mapsto \gamma_x$ of X into $\mathscr{M}_+(X)$ (equipped with the vague topology) is continuous; from this, deduce that one then has $\mathrm{Ch}_{\mathscr{H}}(X) = \check{S}_{\mathscr{H}}(X)$ (note that $\gamma_x = \varepsilon_x$ in $\mathrm{Ch}_{\mathscr{H}}(X)$).

d) In order that every function $f \in \mathscr{C}(S; \mathbf{R})$ be the restriction to S of a function $h \in \mathscr{H}$, it is necessary and sufficient that $\mathscr{H}$ be closed in $\mathscr{C}(X; \mathbf{R})$ and be lattice-ordered.