Folland's Real Analysis Exercise 1.22a The exercise states:
Let $(X, \mathcal{M}, \mu)$ be a measure space, $\mu^*$ the outer measure induced by $\mu$ according to (1.12), $\mathcal{M}^*$ the $\sigma$-algebra of $\mu^*$-measurable sets, and $\bar{\mu}=\mu^* | \mathcal{M}^*$.
a) If $\mu$ is $\sigma$-finite, then $\bar{\mu}$ is the completion of $\mu$. (Use Exercise 18.)
I don't understand what I need to prove in order to show the $\bar{\mu}$ is the completion of $\mu$. The book is not clear to me. Do I need to show that $\mathcal{M}^*$ contains all subsets of measure zero? And that if $E,F \subset \mathcal{M}$ where $\mu(F)=0$, then $\bar{\mu}(E \cup F) = \mu(E)$?
Is that what it means to be the completion of $\mu$?
Thanks for the help.

Hmm. I was going to just point you to the definition in the book, but I see there is no formal definition of the completion of a measure space, instead he proves a theorem and says that that thing in the theorem is the completion.
Do you need to show that $\mathcal M^*$ "contains all subsets of measure zero"? That's not very precisely phrased - I can think of a few things it might mean.
Let $A$ be the set of all $E\subset X$ such that there exists $F\in\mathcal M$ with $E\subset F$ and $\mu(F)=0$. You need to show that $\mathcal M^*$ is equal to the set of all $S\cup E$ with $S\in\mathcal M$ and $E\in A$.
And you need to show that if $E\in M$ and $F\in A$ then $\overline\mu(E\cup F)=\mu(E)$.
