$A, B$ null sets implies $A+B$ is a null set?

Question

Let $A, B\subset \mathbb{R}$ be sets with lebesgue-measure $0$. Define $A+B= \left \{ a+b: a\in A, b\in B \right \}$.

So, is $A+B$ necessarily a null set?

I found this obvious result but nothing more. My intuition tells me that this isn't true, but I don't really manage to approach this problem.

Another question that raises is: is $A+B$ always measurable when $ A, B$ are measurable? Again, no definition of measurability helped me very much.

It's worth mentioning that I know that if A, B are measruable with positive measure, then $A+B$ contains an open interval.

Any suggestions?