A compact in a measurable set The problem is as follows:

Given a Lebesgue-measurable set $M\subset\mathbb{R^n}$ find a compact $C_\epsilon$ and an open $A_\epsilon$, $C_\epsilon\subset M\subset A_\epsilon$, such that $\lambda(A_\epsilon)-\lambda(C_\epsilon)<\epsilon$.

$\lambda$ is the Lebesgue-measure, in particular, if $M$ is not bounded, $k\in\mathbb{N}$, 
\begin{equation}
\lambda(M)=\lim_{k\to+\infty}\lambda(M\cap[-k,k]^n).
\end{equation}
Now, if $M$ is bounded I can find $A_\epsilon$ and $C_\epsilon$ with the required properties.
if $M$ is not bounded but $\lambda(M)<\infty$, how can I find the open and the compact?
Furthermore, if $\lambda(M)=\infty$, 
I think we can not found a compact such that the difference between its measure and the measure of $M$ is small, is it correct?

If $M$ is unbounded and $\lambda(M)<\infty$, start by taking a $k>0$ such $\lambda(M)-\lambda\bigl(M\cap[-k,k]\bigr)<\frac\varepsilon2$.
If $\lambda(M)=\infty$, then you are right: there is no such compact set.
