union bound

Lemma

For any random events $A_1,...,A_k$ (countable):

(here in $\mathrm{Pr}[A_1 \cup A_2 \cup ... \cup A_k]$, $\cup$ means “or”)

Notes

• follows from fact that a measure (such as a probability measure) is $\sigma$-sub-additive, so it holds not only for probability, but for any finite measure

References

1. https://en.wikipedia.org/wiki/Boole's_inequality
2. https://www.probabilitycourse.com/chapter6/6_2_1_union_bound_and_exten.php