Is $\{\omega; \sum_{n=1}^\infty I_{D_n}(\omega)<\infty\}$ a tail event? We have $D_n$ is a sequence of independent events defined on a probability space. Let $E=\{w:\sum_{n=1}^\infty I_{D_n}(w)<\infty\}$. Is $E$ a tail event? Find $P(E)$ if $P(E)>0$.
I know that $\limsup_nD_n$ is a tail event, and it can be written as $\limsup_n D_n=\bigcap_{n=1}^\infty\bigcup_{k=n}^{\infty} D_n=\{\omega:\omega\in D_k \text{ for some } k\geq n\;\;\forall n\}=\{\omega:\omega\in D_n \text{ for infinitely many }n\}$. Thus if only $\limsup_n D_n$ is the tail event, then $E$ is not tail event. But Is $\limsup_n D_n$ the only tail event? What about $\liminf_n D_n$? How can we tell that an event is a tail event or not, without using the sigma field definition? How can we calculate $P(E)$ if it is not tail event? 
 A: Note that $1_{D_n}(\omega) \in \{0,1\}$ for all $n \in \mathbb{N}$ and $\omega \in \Omega$. Therefore
$$\sum_{n=1}^{\infty} 1_{D_n}(\omega)<\infty$$
if, and only if, we can find $N \in \mathbb{N}$ ($N$ may depend on $\omega$) such that for all $n \geq N$
$$1_{D_n}(\omega) = 0,$$
i.e. $\omega \notin D_n$. This means that
$$E = \bigcup_{N \in \mathbb{N}} \underbrace{\bigcap_{n \geq N} D_n^c}_{=:E_N}.$$
Since the sets $E_N$ are increasing (i.e. $E_1 \subseteq E_2 \subseteq \dots$), we have
$$E = \bigcup_{N \geq K} E_N$$
for any (fixed) $K \in \mathbb{N}$. As, by definition, $E_N \in \sigma(D_K, D_{K+1},\dots)$ for all $N \geq K$, this shows
$$E \in \sigma(D_K, D_{K+1},\ldots).$$
Since $K$ is arbitrary, we conclude
$$E \in \bigcap_{K \in \mathbb{N}} \sigma(D_K,D_{K+1},\ldots),$$
i.e. $E$ is a tail event.
A: In fact，you can find that for $\forall\ n\in\mathbb{N}$ $$E=\{\omega;\ \sum_{k=1}^{\infty}I_{D_k}(\omega)<\infty\}=\{\omega;\ \sum_{k=n}^{\infty}I_{D_k}(\omega)<\infty\}$$
Since$$\{\omega;\ \sum_{n=k}^{\infty}I_{D_n}(\omega)<\infty\}\in\sigma(D_n,D_{n+1}\dots)$$
We can conclude that$$E\in\sigma(D_n,D_{n+1}\dots)$$
Then take the intersection we get$$E\in\bigcap_{n=1}^{\infty}\sigma(D_n,D_{n+1}\dots)$$ 
Actually, Durrett's famous book"probability theory and examples" says:
Intuitively, E is a tail event if and only if changing a finite number of values do not affect the occurence of the event.The solution given is just using this idea.
