Let A be a Lebesgue-measurbale subset of $\mathbb{R}^n$. How can I prove that I might exhaust the whole set by let's say cuboids (this is what I mean by exhaustion). I know for open sets that we can find a cover out of closed cuboids with disjoint inner areas.
I thought since every Lebesgue-measurable set is composed of a Borel-set and Null-set we might be able to approximate the set A by open sets which we might cover with countable many cuboids.
But I have no ideawhether that works. What do you think?
This is a consequence of regularity. Given a Lebesgue-measurable set $A$:
If $A$ is bounded, we're done (since our approximating closed set will also be bounded, hence compact).
If $A$ is not bounded, we break it into bounded pieces. For $n\in\mathbb{N}$, let $$A_n=\{a\in A: \Vert a\Vert<n\}$$ be the part of $A$ lying within $n$ of the origin. Given $\epsilon>0$, we iteratively apply regularity to get compact sets $C_n$ such that for each $n$, $C_n\subseteq A_n$ and $\mu(C_n\setminus A_n)<{\epsilon\over 2^{n+1}}$. The family $\{C_n:n\in\mathbb{N}\}$ exhausts $A$ as desired (by countable additivity).