Definition:Decreasing Sequence of Events

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\sequence {A_n}$ be a sequence of events in $\Sigma$.

Then $\sequence {A_n}$ is described as decreasing if and only if:

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

Note

Note that when $\sequence {A_n}$ is considered as a totally ordered set $\struct {A, \subseteq}$, this definition is consistent with the conventional definition of decreasing.

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 a decreasing 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.

Also see

• Definition:Decreasing Sequence of Sets

• Definition:Increasing Sequence of Events
• Definition:Increasing Sequence of Sets

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 1.9$: Probability measures are continuous