I have a simple homework question, but it's not so simple for me...
"Let $S$ be a $\sigma$-algebra on $X$, and $\mu$ a measure on $S$. Assume that $\mu(X)=1$.
Let $\mu^*$ be the outer measure induced by $\mu$. Assume there exists $E\subseteq X$ such that $\mu^*(E)=1$.
Prove that for every $A,B\in S$ such that $A\cap E=B\cap E$, then $\mu(A)=\mu(B)$".
I just can't see how to use $\mu^*(E)=1\ldots$
Thanks in advance!