The algebraic sum of sets of real numbers with strong measure zero sets

摘要

We prove the following theorems: (1) If X has strong measure zero and if Y has strong first category,then their algebraic sum has property s_0. (2) If X has Hurewicz's covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a category set. (3) If X has strong measure zero and Hurewicz's covering property then its algebraic sum with any set in AFJ' is a set in AFJ'. (AFJ' is included in the class of sets always of first category, and includes the class of strong first category sets). These results extend: Fremlin and Miller's theorem that strong measure zero sets having Hurewicz's property have Rothberger's property, Calvin and Miller's theorem that the algebraic sum of a set with the γ-property and of a first category set is first a category set, and Bartoszynski and Judah's characterization of SR~M-sets. They also characterize the property (*) introduced by Gerlits and Nagy is terms of older concepts.