Definition:Limit of Sequence of Events/Decreasing
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Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
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 $\sequence {A_n}_{n \mathop \in \N}$.
From the Elementary Properties of Event Space we have that such a $\ds \bigcap_{i \mathop \in \N} A_i$ is itself an event.
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 1.9$: Probability measures are continuous