Following the idea from Can we swap equivalent events in the conditional part?
I'm wondering if $E[f(Y)|I_{Z=X},X]$ and $E[f(Y)|Z,X]$ match almost surely over $w\in \{Z=X\}$, for any random elements $X,Y,Z$ and real-valued function f.
If this does not hold in general, under what condition does it hold? (e.g. does regular conditional probability or disintegration help?)
This is motivated by the following "intuitive" but unjustified operation:
$P(T(Y)\leq y|Z=X, X=x) = P(T(Y)\leq y|Z=x, X=x)$
(I think in elementary probability courses people tend to do this quite often, but after we introduce the measure-theoretic definition, I'm not sure how to properly describe and justify this using the language from measure theory.)