This question has stumped me.
Assume that $S\subseteq \mathbb{R}$ is meagre and that $S$ has the property that if $x\in S$ and $y$ differs from $x$ in only a finite number of decimal places then $y\in S$. Is it true that $\lambda(S)=0$?
I was thinking that differing by finitely many decimal places is an equivalence relation and noting that if $\lambda(S)>0$ it must be an uncountable union of these equivalence classes, but I could not get this to help me much.