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