Definition:Limit of Sequence of Events/Decreasing

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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.