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.

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.

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

Then:

$\ds N = \bigcup_{i \mathop = 1}^\infty N_i$

As $\mu$ is a measure:

$\map \mu N \ge 0$

Also:

$\ds \map \mu N$

$\le$

$\ds \sum_{n \mathop \in \N} \map \mu {N_n}$

$\ds $

$=$

$\ds \sum_{n \mathop \in \N} 0$

as the $N_n$ are $\mu$-null sets

$\ds $

$=$

$\ds 0$

Hence necessarily:

$\map \mu N = 0$

$\blacksquare$