We consider a class of compact spaces K for which the space P(K) of probabi
lity Radon measures on K has countable tightness in the weak* topology. We
show that that class contains those compact zero-dimensional spaces for whi
ch C(K) is weakly Lindelof, and, under MA + inverted left perpendicular CH,
all compact spaces K with C(K) having property (C) of Corson.