# Null Sets Closed under Countable Union

## Theorem

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

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

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

## Proof

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

 $\ds \mu \left({N}\right)$ $\le$ $\ds \sum_{n \mathop \in \N} \mu \left({N_n}\right)$ Measure is Countably Subadditive $\ds$ $=$ $\ds \sum_{n \mathop \in \N} 0$ The $N_n$ are $\mu$-null sets $\ds$ $=$ $\ds 0$

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

$\blacksquare$