Maximal functions and the additivity of various families of null sets

Steprāns, Juris

doi:10.1090/S0002-9947-2012-05402-X

Maximal functions and the additivity of various families of null sets
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Trans. Amer. Math. Soc. 364 (2012), 3555-3584 Request permission

Abstract:

It is shown to be consistent with set theory that every set of reals of size $\aleph _1$ is null yet there are $\aleph _1$ planes in Euclidean 3-space whose union is not null. Similar results are obtained for circles in the plane as well as other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain maximal operators and a measure-theoretic pigeonhole principle.

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