Completion of sigma fields Let $(\Omega,\mathcal{A})$ be a measurable space. Let $\mu$ be a finite measure.
Define   $\bar{A}^\mu$ as the completion of $\mathcal{A}$ under a measure $\mu$.
I am trying to prove the following exercises:

*

*$B \in \bar{A}^\mu$ if and only if there exists $A$ and $C$ in $\mathcal A$ such that $A\subseteq B\subseteq C$ and $\mu(A \cap C^c) = 0$


*$f:\Omega \to \mathbb{R}$ is $\bar{A}^\mu$ measurable if there exists $g:\Omega \to \mathbb{R}$ which is $\mathcal{A}$ measurable such that $ \begin{equation}
          \mu(f \neq g) = 0
\end{equation}$
I am clueless as to how to proceed. I just know the definition of when a sigma field is called complete. Is there a way to represent explicitly the elements of the completed sigma field? Any hints would be helpful.
 A: "Is there a way to represent explicitly the elements of the completed sigma field?"  Yes, for example your 1. is such a way.  So in this exercise, you are proving it.  There may be other ways as well, but in order to use them, you would have to prove them, and they will not be easier than the proof of 1. proposed here.
Hint: Use the definition of $\bar{\mathcal{A}}^\mu$ which you quoted in a comment:

$\bar{\mathcal{A}}^\mu$ is the smallest sigma field containing $\mathcal A$ that is complete under  $$

So, to work on 1. do this:  First show that the collection
$$
\mathcal C = \{B \subseteq \Omega :\text{there exists $A$ and $C$ in $\mathcal A$ such that $A\subseteq B\subseteq C$ and $\mu(A \cap C^c) = 0$}\}
$$
is a sigma-field, is complete under $\mu$, and contains $\mathcal A$.  That would prove $\bar{\mathcal{A}}^\mu \subseteq \mathcal C$.
Then you would need to prove $\bar{\mathcal{A}}^\mu \supseteq \mathcal C$.

General remark.  This outline is often used when dealing with a definition of the form: "the smallest ... such that ...".  You may have experience in the past  writing proofs using the definition of "least upper bound" which is of that form.
