The Kreps-Yan theorem for $L^{\infty }$

Abstract

We prove the following version of the Kreps-Yan theorem. For any norm closed convex cone $C\subset L^{\infty }$ such that $C\cap L_{+}^{\infty }=\{0\}$ and $C\supset -L_{+}^{\infty }$, there exists a strictly positive continuous linear functional, whose restriction on $C$ is non-positive. The proof uses some tools from convex analysis in contrast to the case of a weakly Lindelöf Banach space, where such approach is not needed.

8 pages