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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:

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

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

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  • $\begingroup$ Just to clarify - here you mean $\lambda(x)\leq\varphi(x)$ for all $x\in X$ ($\lambda-\varphi$ is monotonic increasing rather than just nonnegative)? Also you only require a reference, not a proof? $\endgroup$
    – Ollie
    Commented Jul 25, 2011 at 19:32
  • $\begingroup$ Oops, I mean $\varphi-\lambda$ monotonic increasing! $\endgroup$
    – Ollie
    Commented Jul 25, 2011 at 19:33
  • $\begingroup$ I just want $\varphi - \lambda$ to be non negative. And, no I do not need a proof. $\endgroup$
    – Itaï BEN YAACOV
    Commented Jul 26, 2011 at 8:03

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