Let $X,Y$ be two discrete, integrable random variables, which are defined on a probability space $(\Omega,\mathcal{A},P)$. Recall that the conditional expectation of $Y$ given $X=x$ is defined by $$ E(Y|X=x):=\sum_{y\in Y(\Omega)}yP(Y=y|X=x). $$ Show that $$ E(Y|X)(\omega):=\sum_{x\in X(\Omega)}E(Y|X=x)\chi_{\left\{X=x\right\}}(\omega) $$ defines a discrete random variable
Hello, I have one basic problem. I do not know what I have to show in order to show that $$ E(Y|X)(\omega):=\sum_{x\in X(\Omega)}E(Y|X=x)\chi_{\left\{X=x\right\}}(\omega) $$ defines a discrete random variable.
I think I have to show two things:
(1) $E(Y|X)$ is measurable (but: measurable in which sense?)
(2) $E(Y|X)(\Omega)$ is countable?