I am asked to solve the following exercise: let $A,B$ be Lebesgue measurable subsets of the real line. Prove that $$\lambda(A) + \lambda(B) \leq \lambda(A+B).$$
I am aware of a related question, but that only pertains to finding a counter example so that equality does not hold. A standard example being $A = \mathbb{Z}$ and $B= [0,1],$ with $A+B = \mathbb{R}$.
What I’ve tried: I’ve found that it suffices to assume $A$ and $B$ have finite, nonzero measure, since these cases are trivial. I’ve also found that we may assume that both sets contain $0$, which is probably useful. (The statement for general $A$, $B$ then follows by translation invariance.)
Then I got stuck. Any hints or solutions?
Thanks in advance.