I saw a book where they calculate $E(\tau 1_{\tau<\infty})$ and $E(\tau)$ for some random variable $\tau$ (actually a stopping time of a process). They obtain different results. The problem is that for this variable; $P(\tau<\infty)=1$. Thus in my mind I would expect $E(\tau 1_{\tau<\infty})=E(\tau)$.
Can you give an example of a random variable $\tau$ which satisfies both of the following properties?
1) $P(\tau<\infty)=1$.
2) $E(\tau 1_{\tau<\infty})\neq E(\tau)$