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Geometric cardinal invariants, maximal functions and a measure theoretic pigeonhole principle. (English) Zbl 1105.03049

Summary: 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 will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure-theoretic pigeonhole principle.

MSC:

03E35 Consistency and independence results
03E05 Other combinatorial set theory
03E17 Cardinal characteristics of the continuum
28A25 Integration with respect to measures and other set functions
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