Confusion on integrability of stopping times I know the definition of a $\mathbb F-$stopping time $\tau$ is that for all $n \in\mathbb N $ that $\{ \tau \leq n\} \in \mathcal{F}_{n}$
How do the ideas of integrability and well-definedness of $\tau$ actually fit in to the concept of a stopping. I realize the question is not concise. 
For example, if $P(\tau < \infty)=1$ does this mean that $\tau$ is integrable? I know this is the case if $\tau$ is $-$a.s. bounded, i.e. there exists $c\in \mathbb N$ so that $\tau \leq c-$a.s. and hence $E[\tau] \leq E[c]=c<\infty$

The answer is NO. The sigma fields $\mathcal F_n$ and the fact that $\tau$ is a stopping time are not relevant to the question. You can take $\mathcal F_n$ to be the Borel sigma field of $[0,1]$ for each $n$ and take any non-negative random variable with infinite mean, say $\tau =\sum n I_{(\frac 1 {n+1}, \frac  1 n)}$. 
