Understanding the concept of infinitely often

Question

My book in stochastic processes has this section about i.o which I don’t really understand. Is there any example of any other to explain this? How can I understand this?

It states the following: Let $A_{1}, A_{2}, \ldots$ be a sequence of subsets of $\Omega$. We define $$ \left(A_{n} \text { i.o. }\right)=\bigcap_{n=1}^{\infty} \bigcup_{m=n}^{\infty} A_{m} $$ The abbreviation i.o. stands for infinitely often.

What exactly about the notion of infinitely often is confusing? Be mindful that infinity can have different meanings in different contexts — FShrike, Sep 29 '21 at 17:28
"$(A_n\text{ i.o. })$" strikes me as kind of a weird looking notation. I think I would have written it as $\Phi(A_1,A_2,\ldots)$ (the Greek $\Phi$ looking like a combination of "I" and "o"), or something along those lines. Does anyone know if "$(A_n\text{ i.o. })$" is a standard notation? — Barry Cipra, Sep 29 '21 at 17:39
$(\text{$A_n$ i.o.})$ on its own is not, however ${\text{$A_n$ i.o.}}$ and $\mathbb P(\text{$A_n$ i.o.})$ are both frequently used in probability theory. — bitesizebo, Sep 29 '21 at 17:42
The term is often written as $$\limsup_{n\to \infty} A_n,$$ Never seen th “i.o.” notation before, but Wikipedia includes it. See https://en.wikipedia.org/wiki/Set-theoretic_limit — Thomas Andrews, Sep 29 '21 at 18:12
@Barry Cipra and Thomas Andrews and others interested: Does anyone know if "($A_n$ i.o. )" is a standard notation? --- A notation I like is $\exists^{\infty}$ for "there exist infinitely many" and $\forall^{\infty}$ for "for all but finitely many". I discuss these a bit in my answer to Why haven't mathematicians come up with an efficient way of writing “sufficiently”, e.g. “for $n$ sufficiently large”. — Dave L. Renfro, Sep 29 '21 at 18:14

score 4 · Answer 1 · edited Sep 29 '21 at 18:01

Consider a point $x$. Then the point $x$ is in infinitely many of the sets $A_1, A_2, \ldots$ if and only if there exists no final $A_n$ containing $x$. This means for all $n$ there exists some $m \geq n$ such that $x \in A_m$. This is precisely the case when $$ x \in \bigcap_{n=1}^{\infty} \bigcup_{m = n}^{\infty} A_m.$$ Thus $\bigcap_{n=1}^{\infty} \bigcup_{m = n}^{\infty} A_m$ is exactly the set of points which are in infinitely many of the $A_1, A_2, \ldots$

Understanding the concept of infinitely often

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