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:
 * $$A = \bigcup_{i \in \N} A_i$$

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 $$\bigcup_{i \in \N} A_i$$ 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:
 * $$A = \bigcap_{i \in \N} A_i$$

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 $$\bigcap_{i \in \N} A_i$$ is itself an event.