I have a question on the following problem:
Let $S\subseteq \mathbb{R}$ be a Borel set such that for any $s\in S$, if $s$ and $t$ differ in only a finite number of decimal places then $t\in S$. Then either $\lambda(S)=0$ or $\lambda(\mathbb{R}\setminus S)=0$.
I believe it has something to do with covering theorems, but my explorations in that regard have been unfruitful. I am also confused about the Borel hypothesis. I get the feeling therefore that I am overlooking some useful theorem.
As this is homework, I am looking for direction or advice and not a full answer. Thank you.