Theorem 2

Every real measure on a locally compact space $X$ is the difference of two positive measures.

In view of Theorem 1 and Ch. II, §2, No. 2, Th. 1, it all comes down to proving that a real measure $\mu$ on $X$ is a relatively bounded linear form on the Riesz space $\mathscr{K}(X; \mathbf{R})$. Let $f$ be a continuous function $\geq 0$ on $X$, with compact support $K$; the relation $0 \leq g \leq f$ in $\mathscr{K}(X; \mathbf{R})$ implies that $\|g\| \leq \|f\|$ and that the support of $g$ is contained in $K$. By hypothesis, there exists a number $M_K \geq 0$ such that $|\mu(h)| \leq M_K \cdot \|h\|$ for every function $h \in \mathscr{K}(X, K; \mathbf{R})$; therefore $|\mu(g)| \leq M_K \cdot \|g\| \leq M_K \cdot \|f\|$, which proves the theorem.

The space $\mathscr{M}(X; \mathbf{R})$ of real measures on $X$ is thus identical with the space of relatively bounded linear forms on the Riesz space $\mathscr{K}(X; \mathbf{R})$; we recall that in $\mathscr{M}(X; \mathbf{R})$, the order relation $\mu \leq \nu$ signifies that $\nu - \mu$ is a positive measure, or also that $\mu(f) \leq \nu(f)$ for every function $f \in \mathscr{K}_+(X)$.