Probability of Limit of Sequence of Events
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Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Limit of Increasing Sequence of Events
Let $\sequence {A_n}_{n \mathop \in \N}$ be an increasing sequence of events.
Let $\ds A = \bigcup_{i \mathop \in \N} A_i$ be the limit of $\sequence {A_n}_{n \mathop \in \N}$.
Then:
- $\ds \map \Pr A = \lim_{n \mathop \to \infty} \map \Pr {A_n}$
Limit of Decreasing Sequence of Events
Let $\sequence {B_n}_{n \mathop \in \N}$ be a decreasing sequence of events.
Let $\ds B = \bigcap_{i \mathop \in \N} B_i$ be the limit of $\sequence {B_n}_{n \mathop \in \N}$.
Then:
- $\ds \map \Pr B = \lim_{n \mathop \to \infty} \map \Pr {B_n}$