I'm reading a paper (Residuality of Dynamical Morphisms by Burton, Keane and Serafin) that makes a claim that I've been unable to verify or find a reference for. The claim is made that the extreme points of a compact convex set in a locally convex topological vector space form a $G_\delta$ subset of the space.
I've been able to verify it in the specific context of the paper (sets of invariant measures for a continuous transformation of a compact metric space), but in the article they say a general theorem states that the extreme points of a compact convex set form a $G_\delta$. They don't say whose general theorem! I've looked reasonably hard for a suitable reference without success. Can anyone give me any pointers?