This is question 132Xh from Fremlin Volume 1:
Let $A\subset\mathbb{R}^k$ be non-measurable set with respect to Lebesgue measure $\mu$. Show that there is a bounded measurable set $E$ such that $\mu^{*}(E\cap A)= \mu^{*}(E \setminus A)=\mu(E)>0$. (Hint: take $E^{'}\cap E^{''}\cap B$ where $E'$ is a measurable envelope for $A$, $E^{''}$ is a measurable envelope for $\mathbb{R}^k \setminus A$, and $B$ is a suitable bounded set.)
Here $\mu^{*}(A)=\inf\{\mu(E):E \in \Sigma, A\subset E\}$ is the outer measure induced by $\mu$ for any $A\subset\mathbb{R}^k$. A mesurable envelope for $A$ is a set $E\in \Sigma$ such that $A\subset E$ and $\mu(F\cap E)=\mu^{*}(F\cap A)$ for all $F\in \Sigma$.
I just can't find the right way to choose $B$. Any help is greatly appreciated.