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.
Best Answer
The answer, perhaps surprisingly, is no. You write
But that doesn't exhaust the full variety of the null sets at all. There are quite a lot of very strange null sets out there, and one of the most important examples (and a counterexample to many seemingly-plausible claims) is the existence of a comeager null set - that is, a null set whose complement is the union of countably many nowhere-dense sets. At first glance such a thing may seem impossible, but they do exist - e.g. take the set of all non-absolutely-normal numbers.
(See e.g. the discussion at this MO question. Basically, category and measure are completely orthogonal, although they're both "notions of size," and the interplay between the two (and other notions of size) gives rise to a lot of interesting analysis, topology, and descriptive set theory.)
By the Baire category theorem, comeager null sets can't be covered "efficiently" by $F_\sigma$ sets. Specifically, BCT implies that (in $\mathbb{R}$) meager $G_\delta$ sets are nowhere dense, so comeager $F_\sigma$ sets contain intervals and hence aren't null. So any comeager null $A$ gives a counterexample to your guess.