Definition:Limit of Sequence of Events

Definition

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

Increasing Sequence of Events

Let $\sequence {A_n}_{n \mathop \in \N}$ be an increasing sequence of events.

Then the union:

$\ds A = \bigcup_{i \mathop \in \N} A_i$

of such a sequence is called the limit of the sequence $\sequence {A_n}_{n \mathop \in \N}$.

Decreasing Sequence of Events

Let $\sequence {A_n}_{n \mathop \in \N}$ be an decreasing sequence of events.

Then the intersection:

$\ds A = \bigcap_{i \mathop \in \N} A_i$

of such a sequence is called the limit of the sequence $\sequence {A_n}_{n \mathop \in \N}$.