Theorem 1

Every positive linear form on the Riesz space $\mathscr{K}(X; \mathbf{R})$ is a (positive) real measure on $X$.

For, let $\mu$ be a positive linear form on $\mathscr{K}(X; \mathbf{R})$ and let $K$ be a compact subset of $X$. There exists a continuous mapping $f_0$ of $X$ into $[0, 1]$, with compact support, such that $f_0(x) = 1$ on $K$ (No. 2, Lemma 1). For every function $g \in \mathscr{K}(X, K; \mathbf{R})$, we thus have $-\|g\|f_0 \leq g \leq \|g\|f_0$, consequently $|\mu(g)| \leq \|g\| \cdot \mu(f_0)$, which proves the theorem.

We denote by $\mathscr{M}_+(X)$ the pointed convex cone of positive measures on $X$ (or, what amounts to the same thing, the cone of positive linear forms on the Riesz space $\mathscr{K}(X; \mathbf{R})$).