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...



Extreme points of a compact convex set are a $G_\delta$?
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).


