given a sequence of mutually independent events $\{A_n\}_{n \in \mathbb{N}}$, I am trying to prove that: $$P\left(\bigcap_{i=n}^{\infty}A_i\right) = \lim \limits_{n \to \infty} \prod_{i=n}^{n}P(A_i)$$
It is easy to prove by induction that $P(\bigcap_{i=n}^{m}A_i) = \prod_{i=n}^{m}P(A_i)$ for all $m\in \mathbb{N}$. Now, my problem is due to the limit. In particular:
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Can I write $\bigcap_{i=n}^{\infty}A_i = \lim \limits_{m \to \infty} \bigcap_{i=n}^{m}A_i$ ? If yes, is this the traditional way to interpret intersections –
unions of countable sets? -
Can I write $P (\lim \limits_{m \to \infty} \bigcap_{i=n}^{m}A_i) = \lim \limits_{m \to \infty} P(\bigcap_{i=n}^{m}A_i)$ ? If yes, why and does this property hold in any measurable space? For example, we know that countable additivity is an axiom of measures, but there are no axioms for the one I wrote. So in case it is correct there must be a way to prove it. Maybe applying De Morgan's laws?
Thanks a lot.
Best Answer
In general, if $E_1\supseteq E_2\supseteq\dots$ is a series of nested events, then $P(\bigcap_{i=1}^\infty E_i)=\lim_{n\to\infty}P(E_i)$.
Proof: Let $F_i=E_i-E_{i+1}$ for $i\ge 1$. Then $F_i$ are disjoint and have union $E_1-\bigcap_{i=1}^\infty E_i$. Therefore, \begin{align} P\left(E_1-\bigcap_{i=1}^\infty E_i\right) &=\sum_{i=1}^\infty P(F_i) \\P(E_1)-P\left(\bigcap_{i=1}^\infty E_i\right)&=\lim_{n\to \infty}\sum_{i=1}^n P(F_i) \\&=\lim_{n\to \infty}\sum_{i=1}^n P(E_i)-P(E_{i+1}) \\&\hspace{-.4cm}\stackrel{\text{telescope}}=\lim_{n\to\infty} P(E_1)-P(E_{n+1}) \end{align} THe result then follows by subtracting each side from $P(E_1)$. $\square$
Apply this to the nested series $E_{m}=\bigcap_{i=0}^mA_{n+i}$.