Exercise 10, § 4

a) Let $\mathscr{E}$ be a Riesz space, $\mu$ a positive linear form on $\mathscr{E}$ such that the relation $\mu(|x|) = 0$ implies $x = 0$; $\mu(|x|)$ is then a norm on $\mathscr{E}$, and let us assume that $\mathscr{E}$, equipped with this norm, is complete. It then follows that $\mathscr{E}$ is fully lattice-ordered (Ch. II, §2, Exer. 8 e)). Consequently (Ch. II, §1, Exers. 12 and 13) there exists a locally compact space $X$, the sum of a family $(K_\alpha)$ of compact Stone spaces, such that $\mathscr{E}$ is isomorphic to a space of continuous numerical functions (finite or not) on $X$ that contains the space $\mathscr{H}(X)$. Identifying $\mathscr{E}$ with this space, the restriction of $\mu$ to $\mathscr{H}(X)$ is then a positive measure on $X$. Show that $\mathscr{E}$ is canonically isomorphic to the space $L^1(\mu)$; more precisely, for every $\mu$-integrable function $g$ defined on $X$, there exists one and only one function $f \in \mathscr{E}$ that is equivalent to $g$ for the measure $\mu$ (note that every element $\geq 0$ of $\mathscr{E}$ is the supremum of an increasing sequence of elements of $\mathscr{H}(X)$, and that $\mathscr{H}(X)$ is dense in $\mathscr{L}^1(\mu)$).

b) Deduce from a) that, for every compact space $K$, there exist a locally compact space $S$, the topological sum of a family of compact Stone spaces, and a positive measure $\nu$ on $S$, with support equal to $S$, such that the fully lattice-ordered space $\mathscr{M}(K)$ of measures on $K$, equipped with the norm $\| \mu \|$, is isomorphic to $\mathscr{L}^1(\nu)$ (consider the linear form $\mu \mapsto \mu(K)$ on $\mathscr{M}(K)$).