Universally measurable sets in generic extensions

A subset of a topological space is said to be universally measurable if it is measured by the completion of each countably additive σ-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 universally null sets of real numbers of cardinality א1, and thus that there exist at least 2א1 such sets. Laver showed in the 1970’s that consistently there are just continuum many universally null sets of reals. The question of whether there exist more than continuum many universally measurable sets of reals was asked by Mauldin in 1978. We show that consistently there exist only continuum many universally measurable sets. This result also follows from work of Ciesielski and Pawlikowski on the iterated Sacks model. In the models we consider (forcing extensions by suitably-sized random algebras) every set of reals is universally measurable if and only if it and its complement are unions of ground model continuum many Borel sets. MSC 2010 : 03E35; 28A05 ∗The first author is supported in part by NSF grant DMS-0801009. The research of the second author is supported by NSF grant DMS-0556223. The research of the third author is supported by the United States-Israel Binational Science Foundation. The research in this paper began during a visit by the first author to Rutgers University in October 2008, supported by NSF grant DMS-0600940. Publication no. 947 of second author.