On completed $\sigma$-algebra

Question

I'm having troubles with what seems to be a very simple problem, but I just cannot get my head around formalizing it. Here it is.

• Let $(X,\mathcal{A},\mu)$ be a measured space.
• Let $\mathcal{N}_\mu$ the set of all the null sets ($N\in X$ is a null set if $\exists A\in\mathcal{A} | \mu(A) = 0, N\subset A$).
• Let $\overline{A}^\mu := \{A\cup N | A\in\mathcal{A}, N \in \mathcal{N}_\mu\}$

Prove that $\overline{A}^\mu$ contains $\mathcal{A}$ and $\mathcal{N}_\mu$, then prove that any $\sigma$-algebra containing those two sets contains $\overline{A}^\mu$.

The first part of the problem seems obvious, but I cannot write it. Thanks for your help.

Answer 1

Set $A=\emptyset $ and $N=\emptyset $ respectively.