Probability of Limit of Sequence of Events

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:

$\ds \map \Pr B = \lim_{n \mathop \to \infty} \map \Pr {B_n}$