### 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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