Given two random variables $X,Y$ on the same probability space, given that $E[Y]$ exists, does $E[Y|X=x]$ exist for all $x$ such that $f_X(x)>0$?
My argument : no it does not. If $E[Y|X=x]$ is a Dirac delta pulse at $x=x_0$ with $0<f_X(x_0) < + \infty$ then
$$E[Y] = \int_{\mathbb{R}}^{} E[Y|X=x]f_X(x) dx = f_X(x_0) < +\infty$$
Thus $E[Y]$ exists but $E[Y|X]$ does not.
Is my argument correct? If not how do I go about correcting it?
Definition of Dirac delta pulse taken from : http://mathworld.wolfram.com/DeltaFunction.html