Theorem

Borel zero-one law

If $\{A_n\}$ is a sequence of independent events then $\mathbb{P}(\limsup_{n \to \infty} A_n) = \begin{cases} 0 & \text{iff} \sum_n \mathbb{P}(A_n) \lt \infty \ 1 & \text{iff} \sum_n \mathbb{P}(A_n) = \infty \end{cases}$.