On Definability of Nonmeasurable Sets
Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 653-656
Voir la notice de l'article provenant de la source Cambridge University Press
In [1], Solovay constructed a model of ZFC in which every set of reals in OD (R) is Lebesgue measurable. Here we construct a model in which every equivalence class of sets of reals modulo null sets that is in OD(R) consists of Lebesgue measurable sets. This result immediately implies Solovay's, since the equivalence class of any set of reals in OD (R) is itself OD (R). As a consequence, one cannot provably explicitly define a nonmeasurable set modulo null sets within ZFC. We do not know whether this holds in the model Solovay uses (where an inaccessible cardinal is collapsed to ω 1). Instead, our model is a generic extension of his model.In the model we construct, a somewhat stronger statement holds: every set in OD(R) of sets of reals which has < 2C inequivalent elements modulo null sets, consists entirely of Lebesgue measurable sets of reals.
Friedman, Harvey. On Definability of Nonmeasurable Sets. Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 653-656. doi: 10.4153/CJM-1980-051-3
@article{10_4153_CJM_1980_051_3,
author = {Friedman, Harvey},
title = {On {Definability} of {Nonmeasurable} {Sets}},
journal = {Canadian journal of mathematics},
pages = {653--656},
year = {1980},
volume = {32},
number = {3},
doi = {10.4153/CJM-1980-051-3},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-051-3/}
}
[1] 1. Solovay, R., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Math. 92 (1970), 1–56. Google Scholar
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