Hereditarily additive families in descriptive set theory and Borel measurable multimaps

Hansell, Roger W.

doi:10.1090/S0002-9947-1983-0701521-9

Hereditarily additive families in descriptive set theory and Borel measurable multimaps
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Trans. Amer. Math. Soc. 278 (1983), 725-749 Request permission

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.

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