Definition:Limit of Sequence of Events
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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 $\sequence {A_n}_{n \mathop \in \N}$.