I am interested in studying the quantity $$E[X | X > Y]$$
Where $X$ and $Y$ are any two independent random variables. However, I'm not sure that $E[X | X > Y]$ is always defined. For example, if $X$ is $0$ or $1$ according to a coin flip, and $Y$ is $10$ or $11$ according to a separate independent coin flip, the event $X > Y$ will never occur, and thus I'm not sure if such an expectation could be said to exist.
So my question is, when is $E[X | X > Y]$ well-defined? My best guess is that it could be said to be well-defined if the supports for $X$ and $Y$ "overlap", that is, if $$(\inf(supp(X), \sup(supp(X)) \cap (\inf(supp(Y), \sup(supp(Y)) \neq \emptyset$$ where $supp(X)$ is the support of $X$ and $\inf$ and $\sup$ are infimum and supremum. Note that I'm not requiring the actual supports to share elements, just requiring that their ranges overlap on the number line.
Am I on the right track? Or is $E[X | X > Y]$ always well-defined?