Definition:Limit of Sequence of Events

Definition
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Increasing Sequence of Events
Let $\left \langle{A_n}\right \rangle_{n \in \N}$ be an increasing sequence of events.

Then the union:
 * $\displaystyle A = \bigcup_{i \in \N} A_i$

of such a sequence is called the limit of the sequence $\left \langle{A_n}\right \rangle_{n \in \N}$.

From the definition of event space we have that such a $\displaystyle \bigcup_{i \in \N} A_i$ is itself an event.

Decreasing Sequence of Events
Let $\left \langle{A_n}\right \rangle_{n \in \N}$ be an decreasing sequence of events.

Then the intersection:
 * $\displaystyle A = \bigcap_{i \in \N} A_i$

of such a sequence is called the limit of the sequence $\left \langle{A_n}\right \rangle_{n \in \N}$.

From the Elementary Properties of Event Space we have that such a $\displaystyle \bigcap_{i \in \N} A_i$ is itself an event.