Proposition 3

Let $E$ be a directed ordered vector space, $P$ the set of elements $\geqslant 0$ of $E$, and $x \mapsto M(x)$ a mapping of $P$ into $\mathbf{R}$, with values $\geqslant 0$, such that $M(x + y) = M(x) + M(y)$ for all $x, y$ in $P$. Then, there exists one and only one positive linear form $L$ that extends $M$ to $E$.

Since $E = P - P$, the same reasoning as in Prop. 2 proves, first, the existence and uniqueness of an additive mapping $L$ of $E$ into $\mathbf{R}$ that extends $M$. Proposition 1 then shows that $L(\lambda x) = \lambda L(x)$ for all $\lambda \geqslant 0$ and all $x \in P$, from which it is immediate that $L$ is a linear form.