Boole's Inequality

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

Let $$A_1, A_2, \ldots, A_n$$ be events in $$\Sigma$$.

Then:
 * $$\Pr \left({\bigcup_{i=1}^n A_i}\right) \le \sum_{i=1}^n \Pr \left({A_i}\right)$$

Proof
A direct consequence of the facts that:


 * A Probability Measure is Subadditive;


 * The result Finite Union of Sets in Subadditive Function which gives:
 * $$f \left({\bigcup_{i=1}^n A_i}\right) \le \sum_{i=1}^n f \left({A_i}\right)$$

for a subadditive function $$f$$.

Note
This inequality is also known as Union Bound.