Universally measurable sets in generic extensions

Volume 208 / 2010

Paul Larson, Itay Neeman, Saharon Shelah Fundamenta Mathematicae 208 (2010), 173-192 MSC: Primary 03E35; Secondary 28A05. DOI: 10.4064/fm208-2-4


A subset of a topological space is said to be universally measurable if it is measured by the completion of each countably additive $\sigma$-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 $\aleph_{1}$, 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.


  • Paul LarsonDepartment of Mathematics
    Miami University
    Oxford, OH 45056, U.S.A.
  • Itay NeemanDepartment of Mathematics
    University of California Los Angeles
    Los Angeles, CA 90095-1555, U.S.A.
  • Saharon ShelahThe Hebrew University of Jerusalem
    Einstein Institute of Mathematics
    Edmond J. Safra Campus
    Givat Ram, Jerusalem 91904, Israel
    Department of Mathematics
    Hill Center-Busch Campus
    Rutgers, The State University of New Jersey
    110 Frelinghuysen Road
    Piscataway, NJ 08854-8019, U.S.A.

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