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

Then:

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

Limit of Decreasing Sequence of Events

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

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

Then:

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

Proof

Proof of Limit of Increasing Sequence of Events

Let $\displaystyle B_i = A_i \setminus A_{i-1}$ 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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \Pr \left({A}\right)\)	\(=\)	\(\displaystyle \Pr \left({A_0}\right) + \Pr \left({B_1}\right) + \Pr \left({B_2}\right) + \cdots\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \Pr \left({A_0}\right) + \lim_{n \to \infty} \sum_{k=1}^n \Pr \left({B_k}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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

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

The last sum telescopes.

Hence the result:

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

$\blacksquare$

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.

$\blacksquare$

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.9$: Theorem $1 \ \text{C}$