Universally measurable sets in generic extensions

TL;DR

This paper investigates the size and structure of universally measurable sets in various set-theoretic models, establishing that their number is consistently limited to the continuum, and characterizing their form in certain forcing extensions.

Contribution

It proves that in specific models, the number of universally measurable sets of reals is at most continuum, and characterizes these sets as unions of ground model Borel sets.

Findings

Universally measurable sets are at most continuum in number in certain models.

In some models, universally measurable sets are unions of continuum many Borel sets.

The results connect measure-theoretic properties with set-theoretic forcing extensions.

Abstract

A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $\sigma$-finite Borel measure on the space, and \emph{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 $\aleph_{1}$, and thus that there exist at least $2^{\aleph_{1}}$ 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.