Measure is Subadditive

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

Then $\mu$ is subadditive, that is:
 * $\forall A, B \in \mathcal A: \mu \left({A \cap B}\right) \le \mu \left({A}\right) + \mu \left({B}\right)$

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

So Additive Nowhere Negative Function is Subadditive‎ applies.

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