Probability of Limit of Sequence of Events

Theorem
Let $$\left({\Omega, \Sigma, \Pr}\right)$$ be a probability space.

Limit of Increasing Sequence of Events
Let $$\left \langle{A_n}\right \rangle_{n \in \N}$$ be an increasing sequence of events.

Let $$A = \bigcup_{i \in \N} A_i$$ be the limit of $\left \langle{A_n}\right \rangle_{n \in \N}$.

Then:
 * $$\Pr \left({A}\right) = \lim_{n \to \infty} \Pr \left({A_n}\right)$$

Limit of Decreasing Sequence of Events
Let $$\left \langle{B_n}\right \rangle_{n \in \N}$$ be a decreasing sequence of events.

Let $$B = \bigcap_{i \in \N} B_i$$ be the limit of $\left \langle{B_n}\right \rangle_{n \in \N}$.

Then:
 * $$\Pr \left({B}\right) = \lim_{n \to \infty} \Pr \left({B_n}\right)$$

Proof of Limit of Increasing Sequence of Events
Let $$B_i = A_i \setminus A_{i-1}$$ for $$i \in \N: i > 0$$.

Then:
 * $$A = A_0 \cup B_1 \cup B_2 \cup \cdots$$

is the union of disjoint events in $$\Sigma$$.

By definition of probability measure:

$$ $$

But we have:
 * $$\Pr \left({B_i}\right) = \Pr \left({A_i}\right) - \Pr \left({A_{i-1}}\right)$$ for $$i \in \N: i > 0$$.

So
 * $$\Pr \left({A}\right) = \Pr \left({A_0}\right) + \lim_{n \to \infty} \sum_{k=1}^n \left({\Pr \left({A_i}\right) - \Pr \left({A_{i-1}}\right)}\right)$$

The last sum telescopes.

Hence the result:
 * $$\Pr \left({A}\right) = \lim_{n \to \infty} \Pr \left({A_n}\right)$$

Proof of Limit of Decreasing Sequence of Events
Set $$A_i = \Omega \setminus B_i$$ and then apply De Morgan's laws and the result for an increasing sequence of events.