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**More tales of two \((S)\)-ities.**
*(English)*
Zbl 1108.28003

Let \(X\) be a Polish space and let \(\mathcal P\) denote the family of perfect subsets of \(X\). A set \(M \subseteq X\)

– is Marczewski measurable, or is in \((s)\), if \(\forall P \in \mathcal P\) \(\exists P' \in \mathcal P\) \((P' \subseteq P \wedge ( P' \subseteq M \vee P' \cap M = \emptyset))\);

– is Marczewski null, or is in \((s_{0})\), if \(\forall P \in \mathcal P\) \(\exists P' \in \mathcal P\) \((P' \subseteq P \wedge P' \cap M = \emptyset)\);

– has the Baire property in the restricted sense, or is in \(\mathcal B_{r}\), if \(\forall P \in \mathcal P\), \(P \cap M\) has the Baire property;

– is always of first category, or is in AFC, if \(\forall P \in \mathcal P\), \(P \cap M\) is of the first category;

– is universally measurable, or is in \(\mathcal U\), if \(M\) is measurable with respect to every complete finite Borel measure on \(X\);

– is universally measure zero, or is in \(\mathcal U _{0}\), if \(M\) is has measure zero with respect to every continuous complete finite Borel measure on \(X\).

Using the fact that the Boolean algebra \((s)/(s_{0})\) is complete, the author aims to reprove that there exists an \((s_{0})\) set which is not Lebesgue measurable and does not have the Baire property, and constructs a function between Polish spaces which is not \((s)\)-measurable but its graph is an \((s_{0})\) set.

The author also gives a short alternative proof for two results of S.Baldwin [Real Anal. Exch. 28, No. 2, 415–428 (2003; Zbl 1053.06007)] stating that the Boolean algebras \(\mathcal U / \mathcal U _{0}\) and \(\mathcal B _{r}\)/AFC are not complete. We note that in the proof of the existence of an \((s_{0})\) set which is not Lebesgue measurable and does not have the Baire property there is an unjustifiable use of Fubini and Kuratowski-Ulam theorems. A correction of the proof, based on another result of the paper, was communicated to the reviewer by the author.

– is Marczewski measurable, or is in \((s)\), if \(\forall P \in \mathcal P\) \(\exists P' \in \mathcal P\) \((P' \subseteq P \wedge ( P' \subseteq M \vee P' \cap M = \emptyset))\);

– is Marczewski null, or is in \((s_{0})\), if \(\forall P \in \mathcal P\) \(\exists P' \in \mathcal P\) \((P' \subseteq P \wedge P' \cap M = \emptyset)\);

– has the Baire property in the restricted sense, or is in \(\mathcal B_{r}\), if \(\forall P \in \mathcal P\), \(P \cap M\) has the Baire property;

– is always of first category, or is in AFC, if \(\forall P \in \mathcal P\), \(P \cap M\) is of the first category;

– is universally measurable, or is in \(\mathcal U\), if \(M\) is measurable with respect to every complete finite Borel measure on \(X\);

– is universally measure zero, or is in \(\mathcal U _{0}\), if \(M\) is has measure zero with respect to every continuous complete finite Borel measure on \(X\).

Using the fact that the Boolean algebra \((s)/(s_{0})\) is complete, the author aims to reprove that there exists an \((s_{0})\) set which is not Lebesgue measurable and does not have the Baire property, and constructs a function between Polish spaces which is not \((s)\)-measurable but its graph is an \((s_{0})\) set.

The author also gives a short alternative proof for two results of S.Baldwin [Real Anal. Exch. 28, No. 2, 415–428 (2003; Zbl 1053.06007)] stating that the Boolean algebras \(\mathcal U / \mathcal U _{0}\) and \(\mathcal B _{r}\)/AFC are not complete. We note that in the proof of the existence of an \((s_{0})\) set which is not Lebesgue measurable and does not have the Baire property there is an unjustifiable use of Fubini and Kuratowski-Ulam theorems. A correction of the proof, based on another result of the paper, was communicated to the reviewer by the author.

Reviewer: Tamás Mátrai (Karlsruhe)

### MSC:

28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |

03E15 | Descriptive set theory |

03G05 | Logical aspects of Boolean algebras |