TITLE: Definition of Conditional expectation of Y given X.
QUESTION [2 upvotes]: 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 $E(Y|X=x)$ ?

REPLY [2 votes]: 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.
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