Measure is Subadditive

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

Then $\mu$ is subadditive, that is:
 * $\forall E, F \in \Sigma: \mu \left({E \cup F}\right) \le \mu \left({E}\right) + \mu \left({F}\right)$

Corollary
Let $E_1, \ldots, E_n \in \Sigma$.

Then $\displaystyle \mu \left({\bigcup_{k \mathop = 1}^n E_k}\right) \le \sum_{k \mathop = 1}^n \mu \left({E_k}\right)$.

Proof
A measure is an additive function, and, by definition, nowhere negative.

So Additive Nowhere Negative Function is Subadditive‎ applies.

Hence the result directly:
 * $\mu \left({E \cup F}\right) \le \mu \left({E}\right) + \mu \left({F}\right)$

The corollary is an application of Finite Union of Sets in Subadditive Function.