DEDICATED TO GARRETT BIRKHOFF 1, INTRODUCTION
Let 9'(K) be the collection of all probability measures on a metrizable compact convex set K. Consider a map T: B(K) + B(K) taking each measure p to a measure pT that is more "spread-out" about the same center of mass. A measure on the set E of extreme points is already spread out as far as possible, but suppose that T moves every other measure p. Then we can visualize the possibility that each point measure 6, spreads out towards the extreme set E under the iterations of T, and that S,T" = lim S,T" must be an extreme point measure representing y (having y as its center of mass). Supposing that T is suitably measurable, the limit S = Tm would thus be a measurable spreading out map that is extremal in the above sense. (This would establish the Choquet theorem simultaneously and Bore1 measurably for all the points of K. A short direct proof of this global version of the Choquet theorem has been given by M. Rao [3].)
The type of measurable spreading-out map we are referring to is called a Markov map, and a Markov map that preserves centers of mass is called a dilation. Our simple conjecture above, that if a dilation T moves everything movable then the infinite product S = T* is an extremal dilation, can be shown to be false by an example on the unit interval. However, if T is a dilation that spreads out point measures fur enough, in terms of a suitable scale, then we can prove the theorem.
We shall actually prove a stronger result, by restricitng T to the simplest conceivable type of dilation. Since the theorem cannot be true unless T moves every point measure 6, sitting on a nonextreme point X, the simplest possible value for S,T is a two-point measure US, + bS, centered at X. Thus we shall show that if T is a dilation taking each point mass 6, to a two-point measure aS, + bS, that is spread out at least half 1