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# Additive Families of Low Borel Classes and Borel Measurable Selectors

Published:2010-08-03
Printed: Mar 2011
• J. Spurný,
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Czech Republic
• M. Zelený,
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Czech Republic
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## Abstract

An important conjecture in the theory of Borel sets in non-separable metric spaces is whether any point-countable Borel-additive family in a complete metric space has a $\sigma$-discrete refinement. We confirm the conjecture for point-countable $\mathbf\Pi_3^0$-additive families, thus generalizing results of R. W. Hansell and the first author. We apply this result to the existence of Borel measurable selectors for multivalued mappings of low Borel complexity, thus answering in the affirmative a particular version of a question of J. Kaniewski and R. Pol.
 Keywords: $\sigma$-discrete refinement, Borel-additive family, measurable selection
 MSC Classifications: 54H05 - Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] 54E35 - Metric spaces, metrizability

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