Conditional expectation at a single point

Question

Given two random variables, the conditional expectation $E(Y|X)$ is a $\sigma(X)$ measurable random variable whose integration over any set in $\sigma(X)$ agrees with the integration of $Y$ over the same set.

What does the notation $E(Y | X = x)$ mean? One immediate guess is that it means $E(Y | X) |_{X = x}$. Then for this to be well-defined $E(Y|X)$ must be constant over $\{X = x\}$. Does this definition even make sense cause if $X$ is a continuous random variable then $\{X = x\}$ should be a set of measure zero, then the integral of $E(Y|X)$ over this set is automatically zero, how do we infer the value of $E(Y|X)$ over this set then?

Answer 1

Consider $E[Y|X]$ where $E[|Y|]<\infty$. Then $E[Y|X]=h(X)$ for some Borel $h$. We define $E[Y|X=x]:=h(x)$. For a reference see Klenke's Probability Theory, Corollary 1.97 p. 38 and Definition 8.24 p. 180.