Let $F$ be the set of bijective Borel-measurable functions $f \colon [0,1] \to [0,1]$ that preserve the Lebesgue measure.

>> Is it the case that for every non-Lebesgue-measurable set $A \subset [0,1]$, there exists a countable family $\{f_n\}_{n \in \mathbb{N}} \subset F$ such that $\ \bigcup_{n \in \mathbb{N}} f_n(A)\,$ contains a Lebesgue-measurable set of positive measure?

If so, then there is no *natural* way to extend the Lebesgue measure to include more null sets.

(Conversely, if not, then it seems reasonable to regard all the counterexemplary sets as "kind-of-null sets".)