Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
| (i) | $E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$ |
| (ii) | $E_0, E_1, E_2, \ldots$ is an D1720: Independent event collection in $P$ |
Then
| (1) | \begin{equation} \sum_{n \in \mathbb{N}} \mathbb{P}(E_n) = \infty \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^{\infty} \bigcup_{m = n}^{\infty} E_m \right) = 1 \end{equation} |
| (2) | \begin{equation} \sum_{n \in \mathbb{N}} \mathbb{P}(E_n) < \infty \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^{\infty} \bigcup_{m = n}^{\infty} E_m \right) = 0 \end{equation} |