Lebesgue measure of complement of Q-invariant set

Question

So I was reading up on Linear order of the quotient generated from Vitali relation implies non-measurability of subset of reals. There it was claimed either the relation or it's complement have to have measure 0, with the reasoning that they are invariant under addition of the rationals. I was wondering how to prove this, and if a more general claim were to hold (without choice), namely that for any subset A of the reals with Lebesgue measure >0, (A+Q) will have a null-set complement. It is clear that it has infinite measure, but I can't prove the stronger claim of null-set complement. My approach so far has been trying to show it for Gδ sets (as for open sets the sum will be the whole space) or using the Lebesgue density theorem, but I can't make it work. Any help would be appreciated!