Before exposing the problem we give a series of concepts:
Let $\mathcal{F}$ an algebra of sets and let $\mu_{0}$ a finite and $\sigma$- additive measure.
We denote with $\mathcal{F_\sigma}$ the family all countable unions of sets of $\mathcal{F}$ and with $\mathcal{F_\delta}$ the family all countable intersection of sets of $\mathcal{F}$.
We extend $\mu_{0}$ to two family $\mathcal{F\sigma}$ and $\mathcal{F}_{\delta}$ in the following way:
if $A\in\mathcal{F_{\sigma}}$ \begin{equation} \mu_{1}(A)=\sup\{\mu_{0}(A'), A'\subset A,\; A'\in\mathcal{F}\} \end{equation} else if $B\in\mathcal{F_\delta}$ \begin{equation} \mu_{2}(B)=\inf\{\mu_{0}(B'), B'\supset B,\; B'\in\mathcal{F}\}. \end{equation}
Now, we give the following definition
A set $M\subset X$ is measurable if for all $\epsilon>0$, exist $A\in\mathcal{F_{\sigma}}$ and $B\in\mathcal{F_\delta}$ such that $B\subset M\subset A$ and such that \begin{equation} \mu_{1}(A)-\mu_{2}(B)<\epsilon. \end{equation} The absolute value has been omitted.
The question is: is it reductive to omit tha absolute value?
Thanks!