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Additive families of low Borel classes and Borel measurable selectors. (English) Zbl 1214.54029
It is proved that every point-countable \(\mathcal T\)-additive family in a complete metric space has a \(\sigma\)-discrete refinement for the case \(\mathcal T=F_{\sigma\delta}\). This improves the result of R. W. Hansell [Proc. Am. Math. Soc. 96, 365–371 (1986; Zbl 0584.54035)], who proved it for \(\mathcal T=F_{\sigma}\) and conjectured it might be true for \(\mathcal T\) standing for the class of all Borel sets. The first author proved the above mentioned statement for \(\mathcal T=G_{\delta}\).
The present result follows by a technical construction disproving the \(F_{\sigma\delta}\)-additivity if no \(\sigma\)-discrete refinement exists. It is essentially more complicated than the proof for \(\mathcal T=G_{\delta}\), which followed a similar idea and worked even without point-countability. Here the point-countability is used following former results, respectively methods, by R. Pol [Fundam. Math. 100, 129–143 (1978; Zbl 0389.54021)] and by D. Fremlin [Measure-additive coverings and measurable selectors. Diss. Math. 260, 116 p. (1987; Zbl 0703.28003)].
The result is improved to families in an absolute Suslin space and applied to a selection theorem for separable-valued mappings.

MSC:
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54E50 Complete metric spaces
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