Department of Mathematics, Montanuniversitat, A-8700 Leoben,
Department of Mathematics, Indiana University N. W., Gary, IN 46408,
Abstract. M. Kuczma proved that if A is a set of reals having positive Lebesgue measure then A has a subset A' such that A' has positive Lebesgue measure and is symmetric about a point. We consider similar questions regarding other classes of sets (other then the class of sets having positive Lebesgue measure). In addition, other results and remarks concerning several classes of sets are presented.
1991 Mathematics Subject Classification. 28A05, 39B05.
Key words and phrases. Symmetric sets, Bernstein sets, Hamel bases, positive Lebesgue measure, Baire property, additive functions.