On smítal property

Abstract

Throughout the paper ${\mathbb R}$, will denote the set of real numbers and $\mathbb Q$ - the set of rational numbers. Lebesgue outer (resp. inner) measure on the real line will be denoted by $\lambda^{*}$ (resp. $\lambda_{*}$), whereas $\lambda$ will stand for the Lebesgue measure itself. Moreover, $\mathcal L $ will denote the $\sigma $-algebra of $\lambda $-measurable subsets of $\mathbb R$ and $\mathcal N$ will denote the $\sigma $-ideal of Lebesgue null sets. We will consider a natural topology on $\mathbb R$. Notation $\mathcal B$ will be adopted for the case of subsets of $\mathbb R$ having the Baire property and $\mathcal K$ will denote the $\sigma$-ideal of the sets of the first category. $\mathcal I$$_p$ will denote the $\sigma $-ideal of at most countable sets. The sign $+$ indicates the operation of finding an algebraic sum of two sets $A$ and $B$ contained in $\mathbb R$, so the algebraic sum of this sets will be denoted by $A+B=\left\{a+b: a\in A, b\in B\right\}$. For mathematicians working with Lebesgue measure on the real line, the following lemma become a part of the folklore