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) |
\begin{equation}
\sum_{n \in \mathbb{N}} \mathbb{P}(E_n)
< \infty
\end{equation}
|
Then
| (1) |
\begin{equation}
\mathbb{P} \left( \bigcap_{n = 1}^{\infty} \bigcup_{m = n}^{\infty} E_m \right)
= 0
\end{equation}
|
| (2) |
\begin{equation}
\mathbb{P} \left( \{ \omega \in \Omega : \# \{ n \in \mathbb{N} : \omega \in E_n \} = \infty \} \right)
= 0
\end{equation}
|
