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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.

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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 $\lim_{n\to\infty}P(A_{n+1})=\lim_{n\to\infty}P(A_{n})$

So it would not have been wrong, but more ugly looking.

The lemma is simply saying that for increasing sequences of events, the probability of the union is the limit of the probabilities. For example if you think about random variables this could be:

$P(\{X>0\})=P\big(\bigcup_{n=1}^\infty \big\{X>\frac{1}{n}\big\}\big)=\lim_{n\to\infty}P\big(\big\{X>\frac{1}{n}\big\}\big)$

as $\big\{X>\frac{1}{n}\big\}$ is an increasing sequence of events.

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