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Marczewski-Burstin-like characterizations of \(\sigma\)-algebras, ideals, and measurable functions. (English) Zbl 0940.28002

Summary: \({\mathcal L}\) denotes the Lebesgue measurable subsets of \(\mathbb{R}\) and \({\mathcal L}_0\) denotes the sets of Lebesgue measure \(0\). In 1914 Burstin showed that a set \(M\subseteq \mathbb{R}\) belongs to \({\mathcal L}\) if and only if every perfect \(P\in{\mathcal L}\setminus{\mathcal L}_0\) has a perfect subset \(Q\in{\mathcal L}\setminus{\mathcal L}_0\) which is a subset of or misses \(M\) (a similar statement omitting “is a subset of or” characterizes \({\mathcal L}_0\)). In 1935, Marczewski used similar language to define the \(\sigma\)-algebra \((s)\) which we now call the “Marczewski measurable sets” and the \(\sigma\)-ideal \((s^0)\) which we call the “Marczewski null sets”. \(M\in(s)\) if every perfect set \(P\) has a perfect subset \(Q\) which is a subset of or misses \(M\). \(M\in(s^0)\) if every perfect set \(P\) has a perfect subset \(Q\) which misses \(M\).
In this paper, it is shown that there is a collection \(G\) of \(G_\delta\) sets which can be used to give similar “Marczewski-Burstin-like” characterizations of the collections \(B_{\text{w}}\) (sets with the Baire property in the wide sense) and FC (first category sets). It is shown that no collection of \(F_\sigma\) sets can be used for this purpose. It is then shown that no collection of Borel sets can be used in a similar way to provide Marczewski-Burstin-like characterizations of \(B_{\text{r}}\) (sets with the Baire property in the restricted sense) and AFC (always first category sets). The same is true for \(U\) (universally measurable sets) and \(U_0\) (universal null sets).
Marczewski-Burstin-like characterizations of the classes of measurable functions are also discussed.

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
26A21 Classification of real functions; Baire classification of sets and functions
54E52 Baire category, Baire spaces
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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