Let $X,Y:\Omega \rightarrow \mathbb{R}$ be r.v.
can someone give me an formal definition of this notation:
$$ \mathbb{E}[Y|X=x] $$
is it stil some r.v? Is this just defined if Y is discrete? And how doese it realte to the normal definition of the conditional Expectation:
$E[Y|\sigma(X)]$ is m.b. and $\int_AE[Y|\sigma(X)]dP=\int_AYdP \,\forall A \in \sigma(X)$.
According to the 'normal' definition you have given $E[Y|X]$ or $E[Y|\sigma (X)]$ is measurable w.r.t. $\sigma (X)$ and it can be written as $f(X)$ (up to a set of measure $0$) where $f:\mathbb R \to \mathbb R$ is Borel meaurable. We can define $E[Y|X=x]$ as $f(x)$. This is well-defined if $P[X=x] >0$: If $f(X)=g(X)$ a.s. and $P[X=x] >0$ then $f(x)=g(x)$ necessarily.