Does every null set have a superset which is an $F_{\sigma}$ null set? Let $A$ be a Lebesgue null set in $\mathbb R$. Can we find a set $B$ with the following properties:
1) $A \subset B$
2) $B$ has measure $0$
3) $B$ is an $F_{\sigma}$ set (i.e. a countable union of closed sets).
I suppose so because that null sets I had ever seen have been constructed from finite or countable sets or Cantor set.

A closed set of Lebesgue measure zero has empty interior. So a countable union of such sets, an $F_\sigma$ of measure zero, is meager (also  called "first Baire category), and so are all its subsets. But there are Lebesgue null sets that are not meager, for example the set of those numbers in $[0,1]$ whose binary expansion does not have asymptotically half zeros and half ones (i.e., those numbers whose binary expansions violate the strong law of large numbers).
