Always of the first category sets. (English) Zbl 0571.54025

The paper concerns the \(\sigma\)-ideals of first category sets and of measure zero sets of reals. The main result claims the existence of a first category subset A of \({\mathbb{R}}\) such that \(A+A\) is not of the first category. This results answers a question posed by J. B. Brown and G. V. Cox [Fundam. Math. 121, 143-148 (1984; Zbl 0547.54024)]. The proof uses a theorem of D. Maharam and A. H. Stone on a decomposition of Borel measurable functions, Kuratowski-Ulam theorem on first category sets in products and the concept of always of the first category sets which form a smaller \(\sigma\)-ideal than that of first category sets. The construction of Z in the proof of Theorem 1 is incorrect and the mistake is removed simply in part II of this paper in Proc. 13th Winter School, Rend. Circ. Mat. Palermo, II. Ser., Suppl. 8 (to appear).
Reviewer: P.Holicky


54E52 Baire category, Baire spaces
54C50 Topology of special sets defined by functions
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)


Zbl 0547.54024