Null Sets Closed under Countable Union

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\sequence {N_n}_{n \mathop \in \N}$ be a sequence of $\mu$-null sets.

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

Proof
As $\mu$ is a measure:
 * $\map \mu N \ge 0$

Also:

Hence necessarily:
 * $\map \mu N = 0$

and $N$ is a $\mu$-null set.