Hereditarily additive families in descriptive set theory and Borel measurable multimaps

Author:
Roger W. Hansell

Journal:
Trans. Amer. Math. Soc. **278** (1983), 725-749

MSC:
Primary 54H05; Secondary 04A15, 28A05

MathSciNet review:
701521

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Abstract: A family of Borel subsets of a space is (boundedly) Borel additive if, for some countable ordinal , the union of every subfamily of is a Borel set of class in . A problem which arises frequently in nonseparable descriptive set theory is to find conditions under which this property is "hereditary" in the sense that any selection of a Borel subset from each member of (of uniform bounded class) will again be a Borel additive family. Similar problems arise for other classes of projective sets; in particular, for Souslin sets and their complements. Positive solutions to the problem have previously been obtained by the author and others when is a complete metric space or under additional set-theoretic axioms. We give here a fairly general solution to the problem, without any additional axioms or completeness assumptions, for an abstract "descriptive class" in the setting of generalized metric spaces (e.g., spaces with a -point-finite open base). A typical corollary states that any point-finite (co-) Souslin additive family in (say) a metrizable space is hereditarily (co-) Souslin additive. (There exists a point-countable additive family of subsets of the real line which has a point selection which is not even Souslin additive.) Two structure theorems for "hereditarily additive" families are proven, and these are used to obtain a nonseparable extension of the fundamental measurable selection theorem of Kuratowski and Ryll-Nardzewski, and a complete solution to the problem of Kuratowski on the Borel measurability of complex and product mappings for nonseparable metric spaces.

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DOI:
https://doi.org/10.1090/S0002-9947-1983-0701521-9

Keywords:
Hereditarily-additive families,
descriptive set theory,
Borel measurable multimaps,
measurable selectors,
generalized metric spaces,
point-finite family

Article copyright:
© Copyright 1983
American Mathematical Society