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.

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 1.9$