Universally measurable sets in generic extensions

Paul Larson, Itay Neeman, and Saharon Shelah. "Universally measurable sets in generic extensions." Fundamenta Mathematicae 208.2 (2010): 173-192. <http://eudml.org/doc/286191>.

@article{PaulLarson2010,
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 ℵ₁, and thus that there exist at least $2^\{ℵ₁\}$ 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.},
author = {Paul Larson, Itay Neeman, Saharon Shelah},
journal = {Fundamenta Mathematicae},
keywords = {consistency; sets of real numbers; null sets; universally measurable sets; forcing extensions; random algebras},
language = {eng},
number = {2},
pages = {173-192},
title = {Universally measurable sets in generic extensions},
url = {http://eudml.org/doc/286191},
volume = {208},
year = {2010},
}

TY - JOUR
AU - Paul Larson
AU - Itay Neeman
AU - Saharon Shelah
TI - Universally measurable sets in generic extensions
JO - Fundamenta Mathematicae
PY - 2010
VL - 208
IS - 2
SP - 173
EP - 192
AB - 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 ℵ₁, and thus that there exist at least $2^{ℵ₁}$ 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.
LA - eng
KW - consistency; sets of real numbers; null sets; universally measurable sets; forcing extensions; random algebras
UR - http://eudml.org/doc/286191
ER -