Exercise 9, § 7

Let Y be the topological space whose underlying set is the product $I \times \{-1, 0, 1\}$ in $\mathbf{R}^2$, where $I = [0, 1]$; for every $a \in I$ and every $\varepsilon > 0$, let $U_{a, \varepsilon}$ be the set of $(x, y) \in Y$ such that $|x - a| \leq \varepsilon$ and $(x, y)$ is distinct from $(a, 1)$ and $(a, -1)$; the sets $\{(x, y)\}$ for $y \neq 0$ and $x \in I$ together with the sets $U_{a, \varepsilon}$ form a base for a topology on Y for which Y is a non-metrizable compact space (cf. GT, IX, §2, Exer. 13 d)). Denote by X the topological sum space of Y and a set reduced to a point $\omega$. Let $\mathscr{H}$ be the set of continuous finite numerical functions $h$ on X such that

for all $x \in I$, and $h(\omega) = \int_0^1 h(x,0)\, dx$. Show that $\mathrm{Ch}_{\mathscr{H}}(X)$ consists of the points $(x,y)$ such that $y \neq 0$ and $x \in I$, but that there exists no positive measure $\mu$ of mass 1 on X such that $h(\omega) = \mu(h)$ for all $h \in \mathscr{H}$, and $\mu(X - \mathrm{Ch}_{\mathscr{H}}(X)) = 0$.