Series: Penn State Logic Seminar

Date: Wednesday, July 7, 2004

Time: 11:10 AM - 12:25 PM

Place: 317 Boucke Building

Speaker: Natasha Dobrinen, Penn State, Mathematics


  The von Neumann and Maharam Problems Regarding Measure Algebras,
  part 1

  A measure algebra is (up to isomorphism) just the algebra of
  equivalence classes of some probability measure space, modulo the
  ideal of null sets.  As measure algebras arise so naturally, it is
  interesting to try to characterize measure algebras among Boolean
  algebras.  Von Neumann and Maharam asked whether certain properties
  (the countable chain condition and weak distributivity; and a
  strictly positive Maharam submeasure, respectively) characterize
  measure algebras among Boolean sigma-algebras.  This talk will be
  given in two parts.  Part I will cover basic relevant definitions,
  such as the countable chain condition, weak distributivity, measure,
  and submeasure.  We will state the von Neumann Problem and Maharam's
  Control Measure Problem, as well as some older results.  Several
  examples will be given.  In Part II, we will present some recent
  results of Balcar/Jech/Pazak, Farah/Zapletal, and Velickovic.  In
  particular, we will present the proof of Balcar/Jech/Pazak that it
  is consistent with ZFC that every complete c.c.c., weakly
  distributive Boolean algebra carries a strictly positive Maharam
  submeasure.  This follows from Todorcevic's dichotomy for p-ideals,
  which he proved follows from the Proper Forcing Axiom.