I am not sure I am using the standard definitions so I will open by defining what I need:
Let $X$ be a set, $\nu:\, \mathscr{P}(X)\to[0,\infty]$ will be called an external measure if $\nu(\emptyset)=0$ and for any $\{A_{i}\}_{i=1}^{\infty}\subseteq\mathscr{P}(X)$ (not neccesarily disjoint) it holds that $\nu(\cup_{i=1}^{\infty}A_{i})\leq\sum_{i=1}^{\infty}\nu(A_{i})$
Let $\nu$ be an external measure on a set $X$ then we say that a set $A$ is $\nu$ measurable if for any $E\subseteq X$: $\nu(E)=\nu(E\cap A)+\nu(E\cap A^{c})$
Let $\mu$ be a $\sigma$-additive measure on an algebra $A\subseteq X$, let $\mu^*$ be the outer measure on $X$ that comes from $\mu$.
It was proven in another exercise that the set of $\nu:=\mu^{*}$ measurable sets, $M$, is a $\sigma$ algebra and that $\nu$ is $\sigma$-additive on $M$.
Since $\nu$ is also a measure on the algebra $M$ there is an outer measure $\nu^*=(\mu^*)^{*}$.
The exercise wishes to prove that $(\mu^{*})^{*}=\mu^*$.
I don't really know how to even start here, what I first wanted to figure out is what is $M$. my intuition is that $A=M$ but I tried to prove it and couldn't start (I wrote the definitions but I couldn't see why if $B\in A$ it is $\nu$-measurable, not to mention the other direction which seems even harder).
Can someone please help me get started on this problem ? maybe a hint or an observation that might help me figure out what to do?
ADDED: by an outer measure that comes from a measure I mean $\forall E\subseteq X:\,\nu(E):=Inf\{\sum_{i=1}^{\infty}\mu(A_{i})\,|\, A_{i}\in A,E\subseteq\cup A_{i}\}$