Paul Larson: Universally measurable sets in generic extensions

Week 1, Tuesday June 16, 08:50-09:40

A subset of a topological space is said to be universally measurable if it is measured by the completion of each countably additive finite Borel measure on the space, and universally null if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist uncountable universally null sets, and thus that there exist at least continuum many. Laver showed in the 1970's that consistently there are just continuum many universally null sets. The question of whether there exist more than continuum many universally measurable sets was asked by Mauldin in 1978. We show that consistently there exist only continuum many universally measurable sets. Many interesting questions about the class of universally measurable sets remain open.