probability - Conditional Expectation on Random Variable

In our lecture notes, $X$ and $Y$ are random variable. From what I understand $E[Y|X]$ is an $F_x$ measurable random variable. In our lecture notes the formula for the conditional expectation was written as:

(1) $E[Y|X]=\int_{-\infty}^\infty y \ \ dF_{y|x} \ \ dy$

but I argue that it should be:

(2) $E[Y|X]=\int_{-\infty}^\infty y \ \ dF_{y|x} \ \ dx$

because $E[Y|X]=g(X)$ where $g$ is some function of $x$. Or is it (1) because once we integrate over $y$ we are left with a function only dependent on $x$.

Answer

You are right that the expectation should be some function $g(x)$. To get this equation, you integrate equation (1) over $y$ to get a function that depends on $x$. In other words,
$$E[Y|X] = g(X) = \int_{-\infty}^\infty y dF_{Y|X}.$$