TITLE: Is the measure of the sum equal to the sum of the measures?
QUESTION [9 upvotes]: Let $A,B$ be subsets in $\mathbb{R}$. Is it true that
$$m(A+B)=m(A)+m(B)?$$
Provided that the sum is measurable.
I think it should not be true, but could not find a counterexample.

REPLY [27 votes]: Another example:
$$A=\mathbb{Z},\quad B=[0,1],\quad A+B=\mathbb{R}.
$$