TITLE: Is there an example of $F_{\sigma}$ set in $[0,1]$ with empty interior and with measure $1$?
QUESTION [9 upvotes]: Is there an example of  $F_{\sigma}$ set in $[0,1]$ with empty interior and with measure $1$? I wonder if it exists.

REPLY [5 votes]: Yes. As the commenters have mentioned, it suffices by complementation to construct a dense $G_\delta$ set of measure zero. 
In fact, you can also do this on the entire real line! Let $(q_n)$ be an enumeration of the rationals. Define the set $$E_i = \bigcup_{j\in \mathbb{N}} B(q_j,1/2^{i+j}).$$ Then $E_i$ is a dense open set of Borel measure at most $1/2^i$, and $E_k \subseteq E_i $ for $k>i$. It follows (edit: from the Baire Category Theorem) that $\bigcap_i E_i$ is a dense, $G_\delta$, null set.