Definition:Increasing Sequence of Events

Definition

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

Let $\left \langle {A_n}\right \rangle$ be a sequence of events in $\Sigma$.

Then $\left \langle {A_n}\right \rangle$ is described as increasing iff:

$\forall i \in \N: A_i \subseteq A_{i+1}$

Note

Note that when $\left \langle {A_n}\right \rangle$ is considered as a totally ordered set $\left({A, \subseteq}\right)$, this definition is consistent with the conventional definition of increasing.

Also note that there is a corresponding definition for decreasing sequence of events.

Beware

Note that despite the usual interpretation in natural language of the phrase sequence of events, there is no such assumption that there is any temporal dependency between the events in an increasing sequence of events. That is, they are not necessarily ordered by time. In fact, if you look closely, you will see there is no reference to time in this definition at all.

Sources

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