On uniformly tight sets of probability measures on the rationals

The talk will present results obtained jointly with Piotr Zakrzewski.

  Let P(Q) be the space of probability measures on the rationals Q, equipped with the weak topology. A set A in P(Q) is uniformly tight if for any r > 0 there is a compact set C in Q such that u(C) > 1 - r for every u in A.
 A celebrated theorem of David Preiss asserts that there is a compact set in P(Q) which is not uniformly tight.
  We shall refine this theorem to the followig effect.

   Theorem. There is a Cantor set K in P(Q) such that
(i) the support of any measure in K is locally compact and the supports of any
    pair of distinct measures in K have finite intersection,
(ii) for any Borel map f : K ---> [0,1] there is a Borel set B in K such that B
     is not a countable union of uniformly tight sets and f is either injective
     or constant on B.