Non-lebesgue measurable sets

Question

Find Lebesgue measurable sets $A,B\subset \mathbb{R}^2$ such that $A+B$ is not Lebesgue measurable

How would I go about constructing this? I'm at a lost.

Answer 1

Choose any nonmeasurable set $S\subset \mathbb{R}$. Define $A = S\times\{0\}$ and $B = \{0\}\times S$. Then, $A+B = S\times S$, which is nonmeasurable even though $A$ and $B$ both have measure $0$.

Answer 2

Try solving this yourself first before looking at the full answer. Here's a hint:

All null sets are Lebesgue measurable, but $A$ and $B$ may be null sets while $A+B$ may not be null.

Let $\mathscr{N} \subseteq \mathbb{R}$ be a non-Lebesgue measurable set. Let $A = \mathscr{N}\times \{0\}$ and $B = \{0\}\times [0,1]$. Then $A+B = \mathscr{N}\times [0,1]$ is not Lebesgue measurable. To prove this, it's relatively simple to show that the outer Lebesgue measure of $C\times [0,1]$ for any $C \subseteq \mathbb{R}$ is equal to the outer Lebesgue measure of $C$. Use this to show directly that $A+B$ is not Lebesgue measurable.

edit: fixed a minor mistake