Abstract. In the mid-1960s D. Kazhdan introduced the definition of property T for locally compact groups in terms of their unitary representations. I will describe another approach to this notion which relies on Ergodic Theory and the notion of strong ergodicity. For a countable group G and an action (X;G) of G on a compact metrizable space X, let MG(X) denote the simplex of probability measures on X invariant under G. The natural action of G on the space of functions = f0; 1gG, will be denoted by (;G). I will present two main results. (i) If G has property T then for every G-action the simplex MG(X), when non- empty, is a Bauer simplex (i.e. the set of ergodic measures (extreme points) in MG(X) is closed).(ii) G does not have property T if the simplex MG() is the Poulsen simplex (i.e. the ergodic measures are dense in MG()).Some applications and more recent developmnts will also be indicated.