Construction of the completion of a measure space Let $(X,M, \mu)$ be a measure space. Let $\overline{M}$ be collection of $E \cup Z$ such that $E \in M$ and $Z \subset F$, where $F \in M,$ and $\mu(F) = 0.$ We also know $\bar{\mu}(E) = \mu(E).$
a) Show $\overline{M}$ is smallest sigma algebra containing $M$ and all subsets of elements of $M$ that have measure $0$.
Attempt: Notice if $E_{1} \subset \overline{M}$ then $E_{1} = E \cup Z.$ Since $E, Z \in M$ then $E_{1}^c = (E \cup F)^c \cup (F-Z)$ is in $M$ since $(E \cup F),(F-Z) \in M.$ Hence $\overline{M}$ is sigma algebra. If we let $Z = \emptyset$,then we see $M \subset \overline{M}.$ And since $\emptyset \in M,$ then let $E = \emptyset$, and then all subsets of $M$ with measure $0$ is in $M.$ Does this look okay? I am not sure how to show $\overline{M}$ is the smallest sigma algebra containing $M$
b) Show $\bar{\mu}$ is a measure of $\overline{M},$ and this measure is a completion. 
Attempt: First check $\bar{\mu}$ is a measure of $\overline{M}.$ Notice $\bar{\mu}(\emptyset) = \mu(\emptyset) = 0.$ If $A_{1} \subset A_{2}$, and $A_{1} = B_{1} \cup C_{1}, A_{2} = B_{2} \cup C_{2},$ then $\bar{\mu}(A_1) = \mu(B_1) \le \mu(B_2) = \bar{\mu}(A_2).$ Lastly $\bar{\mu}(\cup_{i=1}^{n}A_{i}) = {\mu}(\cup_{i=1}^{n}B_{i}) \le \sum_{i=1}^{n}\mu(B_i).$ Next for completion, we need to show if $F \in \overline{M}$ and $\bar{\mu}(F) = 0,$ if $E\subset F$ then $E \in \overline{M}.$ I am not sure how to this. 

Attempt of (a):
What you did here seems okay.  Make sure to realize $E_1 \in \overline{M}$, not $E_1 \subset \overline{M}$.  
To show it's the smallest $\sigma$-algebra, you have to show $$\overline{M} = \bigcap N$$ where the intersection is over all complete $\sigma$-algebras on $X$ that contain $M$.
Your question on (b) follows from monotonicity, or write $E$ as $\emptyset \cup E$.
