Universally measurable sets in generic extensions

Paul Larson ¹   ; Itay Neeman ²   ; Saharon Shelah ³

¹ Department of Mathematics Miami University Oxford, OH 45056, U.S.A.
² Department of Mathematics University of California Los Angeles Los Angeles, CA 90095-1555, U.S.A.
³ The Hebrew University of Jerusalem Einstein Institute of Mathematics Edmond J. Safra Campus Givat Ram, Jerusalem 91904, Israel and Department of Mathematics Hill Center-Busch Campus Rutgers, The State University of New Jersey 110 Frelinghuysen Road Piscataway, NJ 08854-8019, U.S.A.

Fundamenta Mathematicae, Tome 208 (2010) no. 2, pp. 173-192

Résumé

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.

Détail

DOI : 10.4064/fm208-2-4

Keywords: subset topological space said universally measurable measured completion each countably additive sigma finite borel measure space universally null has measure zero each atomless measure hausdorff proved there exist universally null sets real numbers cardinality nbsp aleph there exist least aleph sets laver showed consistently there just continuum many universally null sets reals question whether there exist continuum many universally measurable sets reals asked mauldin consistently there exist only continuum many universally measurable sets result follows work ciesielski pawlikowski iterated sacks model models consider forcing extensions suitably sized random algebras every set reals universally measurable only its complement unions ground model continuum many borel sets