It is consistent with ZFC (but not ZFC+CH, of course) that there is a subset $A$ of nonzero outer Lebesgue measure that has cardinality less than $c$. There will then be an extension of Lebesgue measure that assigns non-zero measure to $A$ and there will be a translation-invariant extension of Lebesgue measure that assigns zero measure to $A$ (since there is an extension that assigns zero to all sets of cardinality less than $c$, using the uncountable cofinality of $c$ and the fact that any such set has null inner measure).

Question: Is it consistent with ZFC that there be a *translation-invariant* extension of Lebesgue measure that assigns nonzero measure to some set of cardinality less than $c$?

If yes, then it will be consistent with ZFC that there be a translation-invariant extension of Lebesgue measure which has a set of null measure whose complement has cardinality less than $c$, which will be rather amusing, I think.

(There are two kinds of proofs I've seen of the fact that every set of nonzero Lebesgue measure has cardinality $c$. One kind depends on there being a closed subset of nonzero measure and then a bunch of bisections. That won't work for extensions of Lebesgue measure. The other kind depends on the continuity of convolutions of characteristic functions, which then depends on the $L^1$-continuity of translation, which then depends on approximation by characteristic functions of intervals, and that won't work either.)