TITLE: Extreme points of a compact convex set are a $G_\delta$?
QUESTION [6 upvotes]: Dear All,
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?
Thanks...

REPLY [9 votes]: For a non-metrizable compact convex subset of a locally convex space, extreme points need not even form a Borel set. This has been shown  by Bishop-de Leeuw, The representation of linear functionals by measures on sets of extreme points, Ann.Inst. Fourier (Grenoble) (1959) .
A very good reference for these topics is Phelp's LNM Lectures on the Choquet's theorem (2001).