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Jun 26 at 7:40 comment added David I don't fully understand your notation, but there is a notion of conditional expectation where instead of considering on an event like $\{R \in A \}$ you consider on $R$ directly. Here, $E[f(X) \vert R]$ will still be random but in a sense a function of $R$. If then $X$ has a conditional density with respect to the conditional expectation you write it as an integral. See for example chapter 4.1 of Durrett's book, services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf
Jun 25 at 20:23 history asked Daniel Robert-Nicoud CC BY-SA 4.0