If I have a random variable $X$ and event $A$, I can define new random variables $(X|A)$ and $(X|A^{\complement})$.
Is it well-defined to add these random variables together? That is, $(X|A) + (X|A^{\complement}) = ?$
I think the answer is "no" because conditioning makes each of these random variables a function from effectively different samples spaces, and it is not generally well-defined to add random variables from different sample spaces. In order to add random variables, we would need to know their joint distribution, but it's not clear if it is possible to obtain a joint here.
I was toying with this idea because I was trying to conceive of an interpretation of the expression $X|Y$ as a distribution over conditional random variables. This would make $E[X|Y] = \sum_{y} (X|Y=y)*p_{Y}(y)$. This seemed plausible at first because a distribution over random variables seems like the only mathematical object which can have its expected value taken to result in a single random variable (i.e. $E[X|Y]$ is a single random variable). However, I don't think this summation makes sense - if that is the case, then I'm left concluding that the expression $X|Y$ does not make sense outside of the context of an expected value $E[X|Y]$.