Universally measurable sets in generic extensions

Paul Larson*, Itay Neeman, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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 N1 and thus that there exist at least 2N1 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.

Original languageEnglish
Pages (from-to)173-192
Number of pages20
JournalFundamenta Mathematicae
Volume208
Issue number2
DOIs
StatePublished - 2010

Keywords

  • CPA
  • Random algebra
  • Universally measurable set

Fingerprint

Dive into the research topics of 'Universally measurable sets in generic extensions'. Together they form a unique fingerprint.

Cite this