From this I should also be able to conclude that for sets $B$ with $\mu(B)<\infty$ the existence of an $\mu$-hull follows from the Borel regularity of the measure $\mu $:
I know:
An outer measure $\mu$ is called Borel measure if all open and thus all Borel sets $\mu$-are measurable. $\mu$ is called Borel-regular if in addition each subset $A \subset \mathbb{R}^{N}$ has a Borel superset $B$ with the same measure. $\mu$ is called open-regular if $B$ in the case $\mu(A)<\infty$.
$A \subset \mathbb{R}^{N}$ is so called $\mu$-hull of $B \subset A$ if $A \mu$-measurable, $B \subset A$ and $\mu(T \cap A)=\mu(T \cap B)$ applies to all measurable $T \subset \mathbb{R}^{N}$
But how do I show $\mu(T \cap A)=\mu(T \cap B)$ (or is this analogue to the proof of the question linked above)?