Boole's Inequality

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

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

Then:
 * $\displaystyle \map \Pr {\bigcup_{i \mathop = 1}^n A_i} \le \sum_{i \mathop = 1}^n \map \Pr {A_i}$

Proof
A direct consequence of the facts that:


 * a Probability Measure is Subadditive


 * the result Finite Union of Sets in Subadditive Function which gives:
 * $\displaystyle \map f {\bigcup_{i \mathop = 1}^n A_i} \le \sum_{i \mathop = 1}^n \map f {A_i}$
 * for a subadditive function $f$.

Also known as
This inequality is also known as union bound.