On uniformly tight sets of probability measures on the rationals

Speaker(s)

Roman Pol

Affiliation

University of Warsaw

Date

Oct. 23, 2019, 4:15 p.m.

Room

room 5050

Seminar

Topology and Set Theory Seminar

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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.

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