Measure is Countably Subadditive

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

Then $\mu$ is a countably subadditive function.

Proof
Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of sets in $\Sigma$.

It is required to show that:


 * $\ds \map \mu {\bigcup_{n \mathop \in \N} E_n} \le \sum_{n \mathop \in \N} \map \mu {E_n}$

Now define the sequence $\sequence {F_n}_{n \mathop \in \N}$ in $\Sigma$ by:


 * $F_n := \ds \bigcup_{k \mathop = 0}^n E_n$

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

Hence, $\sequence {F_n}_{n \mathop \in \N}$ is increasing.

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

Now reason as follows:

Hence the result.