I think I have problem with badly written book, or I just can't understand statement.
Let $(X,\mathcal{A},\mu)$ be any measure space and $\mu^*$ be outer and $\mu_*$ inner measure inner measure generated by $\mu$. Then it's true for any $A\subseteq X$:$$\mu_*(A)\leq \mu(A)\leq\mu^*(A)$$
*line under this one is $\mu_*(A)=\mu(A)=\mu^*(A)$ if $A\in\mathcal{A}$
And it's nowhere defined what is $\mu(A)$ for set $A$ that is not from $\mathcal{A}$. What would be definition? How should I "treat" it?
(This inequality is used to prove theorem that A is from completion of $(X,\mathcal{A},\mu)$ if and only if $\mu_*(A)=\mu^*(A)$, so if someone has link to proof f that theorem (since proof from mine book obviously has flaws)I would be grateful =))
The assertion would be correct if instead of $A\subset X$ it would say $A\in\mathcal A$. As you say, the way it is written makes no sense.