See the final line of maths in the segment below, from this paper of Arratia, Goldstein and Gordon 1989.
$s_{\alpha}$ has a $\sigma$ used as part of its definition. This is some sort of notation not defined in the paper. What does it mean? Is it to do with a sum of what is in the brackets?
It’s the sigma algebra generated by the random variables in that set, i.e. $\sigma(X_{\alpha}|\alpha\in J)$ is the sigma-algebra generated by all the $X_{\alpha}$, where $\alpha\in J$.
So the inner expectation is the conditional expectation, given that sigma-algebra.