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 $E(Y|X=x)$ ?
 A: 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.
A: 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*}
