# Trouble understanding the proof and significance of a probability measure being a continuous set function..

From Probability and Random Processes (3rd ed, Grimmett and Stirzaker)

Lemma

Let $A_1,A_2,\ldots$ be an increasing sequence of events so that $A_1 \subseteq A_2 \subseteq A_3 \subseteq \ldots$, and write $A$ for their limit: $$A=\bigcup_{i=1}^{\infty}A_i=\lim_{i\rightarrow\infty}A_i\hspace{0.1em}.$$ Then $\mathbb{P}(A)=\lim_{i\rightarrow\infty} \mathbb{P}(A_i)$.

Proof

$A=A_1 \cup (A_2 \setminus A_1) \cup (A_3 \setminus A_2) \cup \ldots$ is the union of a disjoint family of events. Thus, \begin{align} \mathbb{P}(A)&=\mathbb{P}(A_1)+ \sum_{i=1}^{\infty}\mathbb{P}(A_{i+1}\setminus A_i) \tag1 \\ &= \mathbb{P}(A_1)+\lim_{n\rightarrow\infty}\sum_{i=1}^{n-1}[\mathbb{P}(A_{i+1})-\mathbb{P}(A_i)] \tag2\\ &= \lim_{n\rightarrow\infty}\mathbb{P}(A_n) \tag3 \end{align} $$\tag*{\blacksquare}$$

I understand how writing $A$ as a disjoint union of events gives $(1)$, and how $\mathbb{P}(A_{i+1}\setminus A_i)=\mathbb{P}(A_{i+1})-\mathbb{P}(A_i)$ follows from the properties of a probability space, but I don't understand why we take $n-1$ as the upper limit of summation in $(2)$.

I do see that the partial sum in $(2)$ evaluates to $\sum_{i=1}^{n-1}[\mathbb{P}(A_{i+1})-\mathbb{P}(A_i)]=\mathbb{P}(A_n)-\mathbb{P}(A_1)$, then $(3)$ follows, so it seems like we had to have $n-1$ as the upper limit but I don't quite get what justified us in selecting it.

If I had selected $n$ as the upper limit and ended up with $\lim_{n\rightarrow\infty}\mathbb{P}(A_{n+1})$, would this have been incorrect?

I don't understand what this lemma is saying, what it's significance is or what it means for a set function to be continuous. It seems to me like we are just defining a notation, which I don't think is correct.

You are correct, the authors take the sum to $n-1$ in order to obtain $P(A_n)$. If they selected $n$ they would get $\lim_{n\to\infty}P(A_{n+1})$, but $\lim_{n\to\infty}P(A_{n+1})=\lim_{n\to\infty}P(A_{n})$
$P(\{X>0\})=P\big(\bigcup_{n=1}^\infty \big\{X>\frac{1}{n}\big\}\big)=\lim_{n\to\infty}P\big(\big\{X>\frac{1}{n}\big\}\big)$
as $\big\{X>\frac{1}{n}\big\}$ is an increasing sequence of events.