Proposition 4

— In order that two positive linear forms $L, M$ on a Riesz space $E$ be alien to each other in the space $\Omega$, it is necessary and sufficient that, for every number $\varepsilon > 0$ and every $x \geq 0$ in $E$, there exist two elements $y \geq 0, z \geq 0$ of $E$ such that $x = y + z$ and $L(y) + M(z) \leq \varepsilon$.

Indeed, by the second formula of (3), this condition expresses that $\inf(L, M) = 0$.