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On Borel hull operations. (English) Zbl 1393.54016

Let \(\mathcal M\) denote the \(\sigma\)-ideal of meager subsets of a Polish space \(X\) and let \(\mathcal N\) denote the \(\sigma\)-ideal of the Lebesgue negligible sets in \(X=\mathbb R^n\). For a \(\sigma\)-ideal \(\mathcal I\) of subsets of \(X\) with a Borel base let \(\mathcal S_{\mathcal I}\) denote the \(\sigma\)-algebra on \(X\) generated by \(\mathcal I\) and the family of Borel subsets of \(X\). A Borel hull operation on \(\mathcal F\subseteq\mathcal S_{\mathcal I}\) with respect to \(\mathcal I\) is a mapping \(\psi\) from \(\mathcal F\) to Borel sets such that \(A\subseteq\psi(A)\) and \(\psi(A)\setminus A\in\mathcal I\) for all \(A\in\mathcal F\). It is known and easy to prove that under CH there exist monotone Borel hull operations on \(\mathcal S_{\mathcal M}\) and on \(\mathcal S_{\mathcal N}\). Adding many Cohen reals to a model of CH gives a model with no monotone Borel hull operations on \(\mathcal M\) and on \(\mathcal N\). The authors find some set-theoretic assumptions (for example Martin’s axiom or inequalities between cardinal characteristics of \(\mathcal M\) and \(\mathcal N\)) which imply the non-existence of translation invariant Borel hull operations on \(\mathcal M\) or on \(\mathcal N\). They also prove that the existence of a monotone hull operation on \(\mathcal M\) with respect to \(\mathcal M\) is equivalent to the existence of such a hull operation on \(\mathcal S_{\mathcal M}\).

MSC:

54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
03E15 Descriptive set theory
03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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