Null Sets Closed under Countable Union

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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:

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

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

$\blacksquare$


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