Can a Measurable set be written as union of rectangles?

Question

Let $A \subset \mathbb{R}^n$ a Lebesgue measurable set. My question is: can $A$ be written as countable union of (closed) rectangles? i.e. $\exists \, Q_k \subset \mathbb{R}^n , k\in \mathbb{N}, $ (closed) rectangles such that \begin{align*} A = \cup_{k=1}^{\infty} Q_k \end{align*}

Thanks!!

Answer 1

The set $\mathbb I$ of irrationals in $\mathbb R$ is an example. It's Lebesgue measurable, is uncountable, and has no interior. Thus no closed interval except for a singleton can lie in $\mathbb I,$ and there are uncountably many of these.