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Combinatorics on \(\sigma\)-algebras and a problem of Banach. (English) Zbl 0599.28003

Certain problems related to a problem posed by Banach on measurable \(\sigma\)-algebras are addressed. For example, under certain set-theoretic assumptions (existence of a \(2^{\omega}\) scale, uniformity of Lebesgue measure), for any \(0<\kappa \leq \omega\), a collection of uniformly measurable \(\sigma\)-algebras is obtained such that any union of \(<\kappa\) of them generate a uniformly measurable \(\sigma\)-algebra and any union of \(\geq \kappa\) generate a nonmeasurable \(\sigma\)-algebra.
Reviewer: J.W.Hagood

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A10 Real- or complex-valued set functions
03E15 Descriptive set theory
03E50 Continuum hypothesis and Martin’s axiom
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