# Definition:Decreasing Sequence of Events

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## 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

## Sources

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