Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A family $ \mathcal{B}$ of Borel subsets of a space $ X$ is (boundedly) Borel additive if, for some countable ordinal $ \alpha $, the union of every subfamily of $ \mathcal{B}$ is a Borel set of class $ \alpha $ in $ X$. 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 $ \mathcal{B}$ (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 $ X$ 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 $ \sigma $-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 $ {F_\sigma }$ 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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54H05, 04A15, 28A05

Retrieve articles in all journals with MSC: 54H05, 04A15, 28A05

Additional Information

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