6533b852fe1ef96bd12ab87b

RESEARCH PRODUCT

A Note on Algebraic Sums of Subsets of the Real Line

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AbstractWe investigate the algebraic sums of sets for a large class of invari-ant ˙-ideals and ˙- elds of subsets of the real line. We give a simpleexample of two Borel subsets of the real line such that its algebraicsum is not a Borel set. Next we show a similar result to Proposition 2from A. Kharazishvili paper [4]. Our results are obtained for ideals withcoanalytical bases. 1 Introduction We shall work in ZFC set theory. By !we denote natural numbers. By 4wedenote the symmetric di erence of sets. The cardinality of a set Xwe denoteby jXj. By R we denote the real line and by Q we denote rational numbers. IfAand Bare subsets of R n and b2R , then A+B= fa+b: a2A^b2Bgand A+ b= A+ fbg. Similarly, if AR, BR n and a2R, then AB=fab: a2A^b2Bgand aB= fagB.We say that a family Fof subsets of R is invariant if for each A2F, a2Qand b2R we have aA+ b2F(see [3]).Let Ebe a polish space. If x2Eand ">0, then by B(x;") we denote theball with center xand radius ". The family of Borel subsets of Ewe denoteby Bor(E). For each <!