Exercise 16, § 4

Let $X$ be a completely regular space, and let $\mathscr{C}(X)$ (resp. $\mathscr{C}^b(X)$) be the Riesz space of continuous (resp. continuous and bounded) numerical functions on $X$.

a) Show that if a linear form $\lambda$ on $\mathscr{C}(X)$ is continuous for the topology of compact convergence, it is relatively bounded.

b) Let $\lambda$ be a positive linear form on $\mathscr{C}(X)$. Show that if $\lambda$ is zero on $\mathscr{C}^b(X)$, then it is zero on $\mathscr{C}(X)$ (let $\varphi$ be the canonical mapping of $\mathscr{C}(X)$ onto the quotient Riesz space $\mathscr{C}(X)/\mathscr{C}^b(X)$ (Ch. II, §1, Exer. 4); show that if $h$ is a continuous function $\geqslant 0$ and is not bounded on $X$, then $n \varphi(h) \leqslant \varphi(h^2)$ for every integer $n > 0$).

c) Let $\beta X$ be the Stone–Čech compactification of $X$, the compact space obtained by equipping $X$ with the coarsest uniform structure for which the functions in $\mathscr{C}^b(X)$ are uniformly continuous and by completing the uniform space so obtained; every function $f \in \mathscr{C}(X)$ may then be extended by continuity to a continuous function $\widetilde{f}$ on $\beta X$, with possibly infinite values (consider $f/(1+|f|)$). If $\lambda$ is a positive linear form on $\mathscr{C}(X)$, then the restriction of $\lambda$ to $\mathscr{C}^b(X)$ is of the form $f \mapsto \mu(\widetilde{f})$, where $\mu$ is a positive measure on $\beta X$; show that for every function $f \in \mathscr{C}(X)$, $\widetilde{f}$ is integrable for $\mu$ and $\lambda(f) = \mu(\widetilde{f})$ (make use of $b$), noting that every function $\geqslant 0$ belonging to $\mathscr{C}(X)$ is the upper envelope of a sequence of functions in $\mathscr{C}^b(X)$.

d) Show that every point $x_0$ of the support of $\mu$ that does not belong to $X$ has the following property: for every decreasing sequence $(V_n)$ of neighborhoods of $x_0$, the intersection of the $V_n$ contains at least one point of $X$. Converse.

e) Deduce from d) that if $X$ is locally compact and countable at infinity, then the support of $\mu$ is contained in $X$ (compare with h)).

f) If $X$ is discrete, show that the support of $\mu$ is finite (in the contrary case, form a function $f \geqslant 0$ defined on $X$ such that $\widetilde{f}$ is not $\mu$-integrable).

g) Show that if a linear form $\lambda$ on $\mathscr{C}(X)$ is positive and continuous for the topology of compact convergence, then the support of $\mu$ is contained in $X$ (cf. Ch. III, §2, No. 3, Prop. 11).

h) Let $X_0$ be the compact space obtained by adjoining a point at infinity $\omega$ to the locally compact space $X$ defined in GT, I, §9, Exer. 11, let $Y$ be the subspace of $\overline{\mathbf{R}}$ formed by the integers $\geqslant 0$ and $+\infty$, and let $Z$ be the locally compact space complementary to the point $(\omega, +\infty)$ in the product space $X_0 \times Y$. Show that every continuous numerical function on $Z$ is bounded and that $f(z)$ tends to a limit as $z$ tends to the point at infinity $(\omega, +\infty)$ of $Z$. From this, deduce that if $\mu$ is the measure on $Z$ defined by the mass $1/2^n$ placed at the point $(\omega, n)$ for every $n \geqslant 0$, then every continuous function on $Z$ is $\mu$-integrable, but the support of $\mu$ is not compact.