A dichotomy for the convex spaces of probability measures
  We show that every nonempty compact and convex space M of probability Radon measures either contains a measure which has `small' local character in M or else M contains a measure of `large' Maharam type. Such a dichotomy is related to several results on Radon measures on compact spaces and to some properties of Banach spaces of continuous functions. 
Miko{\l}aj Krupski and Grzegorz Plebanek
math.FA