Probability Measure is Subadditive

Theorem
Let $$\left({\Omega, \Sigma, \Pr}\right)$$ be a probability space.

Then $$\Pr$$ is a subadditive function.

Proof
By definition, a probability measure is a measure.

So it is additive function and nowhere negative.

So Additive Nowhere Negative Function is Subadditive‎ applies.

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