Nonmeasurable algebraic sums of sets of reals

We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y . Some additional related questions concerning measure, category and the algebra of Borel sets are also studied. Sierpiński showed in [14] that there exist two sets X, Y ⊆ R of Lebesgue measure zero such that their algebraic sum, i.e. the set X + Y = {x + y : x ∈ X, y ∈ Y } is nonmeasurable. The analogous result is also true for the Baire property. Sierpiński’s construction has been generalized to other σ-algebras and σ-ideals of subsets of R. Kharazishvili proves in [10] that for every σ-ideal I which is not closed under algebraic sums and every σ-algebra A such that the quotient algebra A/I satisfies the countable chain condition, there exist sets X,Y ∈ I such that X + Y 6∈ A. A similar result was proved by Cichoń and Jasiński in [3] for every σ-ideal I with coanalytic base and the algebra Bor[I] (i.e. the smallest algebra containing I and Bor). Ciesielski, Fejzić and Freiling prove in [4] a stronger version of Sierpiński’s theorem. They show that for every set C ⊆ R such that C + C has positive outer measure there exists X ⊆ C such that X + X is not Lebesgue measurable. In particular, starting with such a set C of measure zero (the “middle third” Cantor set in [0, 1] for example), we obtain Sierpiński’s example as a corollary. In the first section our paper we introduce an elementary notion of the Perfect Set Property of pairs 〈I,A〉, where I is a σ-ideal of subsets of R and A ⊇ I is any family of subsets of R. Using a simple argument, we generalize the results of Sierpiński, Cichoń–Jasiński and Ciesielski–Fejzić–Freiling onto pairs with the Perfect Set Property. The main result of the second section is a stronger version of this theorem for measure and category. Namely, we show that if C is a measurable set such that C + C does not have measure zero, then we can find a measure zero set X ⊆ C such that X +X is nonmeasurable. The analogue for Baire category is also proved. In section 3 similar questions concerning the algebra of Borel sets are studied. Although it is known that this algebra is not closed under taking algebraic sums, we show that there exists an uncountable Borel set P ⊆ R such that for every pair of Borel sets A,B ⊆ P the set A + B is Borel. Standard set-theoretic notation and terminology is used throughout the paper. The reader may check [1] or [9] for basic definitions. We work in the space R (as an additive group, with Lebesgue measure). The arguments of the first section can be easily generalized to Polish groups which have a structure of a linear space over a countable field. In particular, they remain valid 2000 Mathematics Subject Classification. Primary: 28A05, 03E15; secondary 54H05.