I am confused with the meaning of $E[Y^2|X]$ for which $E[Y|X]$ exists. When there is a density for $E[Y|X=x]$ I know that it would simply be $E[Y^2|X=x] = \int y^2 p_{Y|X=x}dy$.
But what about the general case where $Y$ is the random variable that satisfies $Y=E[X|\mathscr{G}]$ is $\mathscr{G}$ measurable and we have the partial averaging property $$\int_A Y dP = \int_A E[Y|X] dP = \int_A X dP$$ for all $A\in \mathscr{G}$?
To sum it up, I am wondering how to interpret $E[Y^2|X]$ if $E[Y|X]$ exists in the standard form but without a density and its relation to $E[Y|X]$.