02 - Properties of Probability Measures

Continuity of Probability from Below lemma

Let $A_n$ for $n\ge 1$ be an increasing sequence of events with

$$A_1\subseteq A_2\subseteq A_3\subseteq \cdots$$

Then

$$\underset{n\to \infty }{\lim }\mathbb{P}(A_n)=\mathbb{P}\left(\bigcup _{n\ge 1}{A}_n\right)$$

Proof

As sequence $A_n$ is increasing then

$$\begin{array}{rl}\mathbb{P}\left(\bigcup _{n\ge 1}{A}_n\right)& =\mathbb{P}\left(A_1\cup \bigcup _{n\ge 1}(A_{n+1}\setminus A_n)\right)\\ & =\mathbb{P}(A_1)+\sum _{i=1}^{\infty }\mathbb{P}(A_{i+1}\setminus A_i)\text{(using countable additivity)}\\ & =\mathbb{P}(A_1)+\underset{n\to \infty }{\lim }\sum _{i=1}^{n-1}\mathbb{P}(A_{i+1}\setminus A_i)\\ & =\underset{n\to \infty }{\lim }\left(\mathbb{P}(A_1)+\sum _{i=1}^{n-1}\mathbb{P}(A_{i+1}\setminus A_i)\right)\\ & =\underset{n\to \infty }{\lim }\mathbb{P}\left(A_1\cup \bigcup _{1\le i\le n-1}(A_{i+1}\setminus A_i)\right)\\ & =\underset{n\to \infty }{\lim }\mathbb{P}(A_n)\end{array}$$