Does there exist a measurable subset $A \subset \mathbb{R}$, such that $\mu(A)$ is finite, but $\mu(\{a+b|a,b\in A\}) = \infty$? Here $\mu$ stands for Lebesgue measure.

If such subset exists, it can not be bounded:
Suppose it is. Then there exists such $a \in \mathbb{R}$ such that $A \subset [-a;a]$. That results in $\{a+b|a,b\in A\} \subset [-2a;2a]$, from which follows that $\mu(\{a+b|a,b\in A\}) \leq 4a$ is finite.

However, I do not know, how to solve the problem in general.

Any help will be appreciated.