## Theorem

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

Then $\mu$ is a countably subadditive function.

## Proof

Let $\left({E_n}\right)_{n \in \N}$ be a sequence of sets in $\Sigma$.

It is required to show that:

$\displaystyle \mu \left({\bigcup_{n \mathop \in \N} E_n}\right) \le \sum_{n \mathop \in \N} \mu \left({E_n}\right)$

Now define the sequence $\left({F_n}\right)_{n\in\N}$ in $\Sigma$ by:

$F_n := \displaystyle \bigcup_{k \mathop = 1}^n E_n$

By Subset of Union, it follows that, for all $n \in \N$, $F_n \subseteq F_{n+1}$.

Hence, $\left({F_n}\right)_{n\in\N}$ is increasing.

It is immediate that $F_n \uparrow \displaystyle \bigcup_{n \mathop \in \N} E_n$, where $\uparrow$ signifies the limit of an increasing sequence of sets.

Now reason as follows:

 $\displaystyle \mu \left({\bigcup_{n \mathop \in \N} E_n}\right)$ $=$ $\displaystyle \lim_{n \to \infty} \mu \left({F_n}\right)$ Characterization of Measures, $(3)$ $\displaystyle$ $=$ $\displaystyle \lim_{n \to \infty} \mu \left({E_1 \cup \cdots \cup E_n}\right)$ Definition of $F_n$ $\displaystyle$ $\le$ $\displaystyle \lim_{n \to \infty} \sum_{k \mathop = 1}^n \mu \left({E_k}\right)$ Measure is Subadditive: Corollary $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop \in \N} \mu \left({E_k}\right)$

Hence the result.

$\blacksquare$