Definition:Limit of Sequence of Events/Increasing

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

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 $\sequence {A_n}_{n \mathop \in \N}$.

From the definition of event space we have that such a $\ds \bigcup_{i \mathop \in \N} A_i$ is itself an event.