# 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:

$\displaystyle \Pr \left({\bigcup_{i \mathop = 1}^n A_i}\right) \le \sum_{i \mathop = 1}^n \Pr \left({A_i}\right)$

## Proof

A direct consequence of the facts that:

$\displaystyle f \left({\bigcup_{i \mathop = 1}^n A_i}\right) \le \sum_{i \mathop = 1}^n f \left({A_i}\right)$

for a subadditive function $f$.

$\blacksquare$

## Also known as

This inequality is also known as Union Bound.

## Source of Name

This entry was named for George Boole.