Null Sets Closed under Countable Union

Theorem
Let $\left({X, \mathcal A, \mu}\right)$ be a measure space.

Let $\left({N_n}\right)_{n \in \N}$ be a sequence of $\mu$-null sets.

Then $N := \displaystyle \bigcup_{n \in \N} N_n$ is also a $\mu$-null set.

Proof
As $\mu$ is a measure, $\mu \left({N}\right) \ge 0$. Also:

Hence necessarily $\mu \left({N}\right) = 0$, and $N$ is a $\mu$-null set.