Hi there,

I'm working through a basic proof in a probability theory book, and I'm stuck trying to work out whether one of the statements is a typo, or I'm missing something:

$\displaystyle A_k$ is an arbitrary indexed set

If

$\displaystyle \omega\in\limsup_{n\rightarrow\infty}=\bigcap_{n=1 }^{\infty}\bigcup_{k=n}^{\infty}A_k$

then since for every $\displaystyle n$, $\displaystyle \omega\in\cup_{k\ge{n}}A_k$, for all $\displaystyle n$ there exists some $\displaystyle k_n\ge{n}$ such that $\displaystyle \omega\in{A_k}_n$ and therefore

$\displaystyle \sum_{j=1}^{\infty}1_A_{j}(\omega)\ge\sum_{n}{1_{{ A_k}_n}}(\omega)=\infty$

And at this point I'm confused: it seems to be that by construction, the indicator function in the second sum should always return 1, but the first does not necessarily. Therefore while they are both infinite sums, the first one may be smaller than the second, not the other way around.