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

Question

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.

Answer 1

Interesting enough you can even find such a set of measure $0$.

Indeed, if $C$ is the tertiary Cantor set then $C+C=\{ a+b : a,b \in C\} =[0,2]$ see this question

Then $$A= \bigcup_{n \in \mathbb Z} 2n+C $$ has measure $0$ but $A+A=\mathbb R$.

Answer 2

$A=[0,1]\cup\mathbb Z$ is a closed set, $\mu(A)=1$, and $\{a+b:a,b\in A\}=\mathbb R.$