Universally measurable sets in generic extensions

by Larson and Neeman and Shelah. [LrNeSh:947]
Fundamenta Math, 2010
A subset of a topological space is said to be universally measurable if it is measurable with respect to every complete, countably additive sigma-finite measure on the space, and universally null if it has measure zero for each such atomless measure. In 1934, Hausdorff proved that there exist universally null sets of cardinality aleph_1, and thus that there exist a least 2^{aleph_1} such sets. Laver showed in the 1970's that consistently there are just continuum many universally null sets. The question of whether there exist more than continuum many universally measurable sets was asked by Mauldin no later than 1984. We show that consistently there exist only continuum many universally measurable sets.

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