The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)

Let $X$ be an ordered vector space over the reals, and let $\varphi\colon X \to {\mathbf R}$. Then the following are equivalent:

There exists a positive linear functional $\lambda\colon X \to {\mathbf R}$ with $\lambda \leq \varphi$.

Whenever $\sum x_i \geq 0$, also $\sum \varphi(x_i) \geq 0$.