Definition of Conditional expectation of Y given X.

Let $(S,F,P)$ be probability space.

Let $X,Y$ be continuous random variables from $S$ to $\mathbb{R}.$

Formal Definition of Conditional Expectation $E(Y|X)$ of $Y$ to X is

$\sigma{(X)} - Borel$ measurable function such that $$ \int_A E(Y|X) dP = \int_A Y dP$$

for all $A \in \sigma(X)$ where $\sigma(X)$ is sigma algebra generated $\{X^{-1}(B) :$ $B$ is borel set$\}$

What is the definition of $E(Y|X=x)$ ?

One can also define $\mathbb{E}(Y|X=x)$ through the factorization lemma: Since $Z = \mathbb{E}(Y|X)$ is $\sigma(X)$-measurable, there is some measurable $g:\mathbb{R} \to \mathbb{R}$ that is unique on $X(\Omega)$ such that $Z = g\circ X$. Now we can define $\mathbb{E}(Y|X=x) = g(x)$. Note that this depends on the version $Z$ of $\mathbb{E}(Y|X)$ that one takes.

This is defined in many probability books, for example, see Shiryaev. Specifically, \begin{align*} m(x) \equiv E(Y \mid X=x) \end{align*} is a Borel measurable function such that, for any Borel measurable set $A$, \begin{align*} \int_{\{X \in A\}} Y dP &= \int_A m(x) P_{X}(dx), \end{align*} where $P_X(dx)$ is the Lebesgue-Stieltjes measure generated by the distribution function of $X$, that is, for any Borel measurable set $B$, \begin{align*} P_X(B) = P(X \in B). \end{align*} It can also be shown that (see Page 196 of Shiryaev), \begin{align*} \int_A m(x) P_{X}(dx) = \int_{\{X \in A\}} m(X) dP. \end{align*} In other words, \begin{align*} m(X) = E( Y \mid X). \end{align*}