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 Title: The von Neumann and Maharam Problems Regarding Measure Algebras, part 1 Abstract: 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.