Is there a standard general definition of expectation $\mathbb{E}(X_\tau)$ for a stopping time $\tau$ and a sequence of random variables $\{X_n\}$? Usually the assumption $\mathbb{P}(\tau<\infty)=1$ is made, so to use $\mathbb{E}\sum_{n=1}^\infty X_n\mathbb{1}_{\{\tau=n\}}$. However it is not clear what the definition should actually be for a stopping time that has a nonzero chance to never stop.
I see two possibilities, but I do not know whether any of them is standard (or perhaps both of them are, depending on the context). Clearly the $\limsup X_n$ is always an option, so perhaps one could just add $\mathbb{E}\limsup X_n \mathbb{1}_{\{\tau=\infty\}}$ to the previous definition. Otherwise we could conventionally define $X_\infty$ to be some arbitrary value and add $\mathbb{E} X_\infty \mathbb{1}_{\{\tau=\infty\}}$.
Both options seem rather conventional and perhaps there is not a natural way of doing this, because in a sense we are compactifying the problem, and this is perhaps not natural per se. Does anybody know if any of them (or others that are better and I haven't thought of yet) is preferable or more standard?