On the econometric notation of conditional expectation. I have recently studied probability theory and I know the definition of conditional expectation, going back on some econometric notes I am unsure on some notation.
Say we have $X,Y$ random variables defined on a probability space then my econometric notes say:
$$E[X| Y] = \hat{X} = \int x P(x|Y) \,dx$$ 
What does the notation $P(x|Y)$ stand for?
Does it refers to the probability density of the conditional expectation ? it can't be he conditional expectation of the indicator function of the event $X^{-1}(x)$, correct?
Now say we have another AC r.v. $V$ with density $P_V$ and that the relation $X = Y+V$ holds
then the econometric notes say 
$$\int xP(x|Y) \,dx = \int x P_V ( x- y) \,dx $$
I don't understand how a change of density took place, have we used Radon-Nikodim? Could somebody provide some rigor to these manipulations? 

I believe that $P(x | Y)$ is a random probability density function whose value is (or becomes) $P(X = x | Y = y)$ when $Y = y$. This intuitive explanation suffices to understand the integral.
Put differently (if additional formalism helps), $P(x | Y)$ is a function from outcomes involving $Y$ to probability density functions over $X$. This can be written as $P : Y \rightarrow (\wp(\Omega) \rightarrow [0,1])$.
Your text is playing a bit fast and loose with notation, but if we have $X = Y + V \Rightarrow V = X - Y$, and if $Y = y$ is known, then we can apply a form of convolution of probability density functions. The key insight is that $P(X = x, Y = y) = P(Y = y, X - Y = x - y)$ (you can convince yourself of this using a set-theoretic argument with diagrams), from which we have $P(X - Y = x - y | Y = y) P(Y = y)$ by the definition of conditional probability.
In this case, we can write:
$$P(X = x | Y = y) = \frac{P(X = x, Y = y)}{P(Y = y)} = \frac{P(X - Y = x - y | Y = y) P(Y = y)}{P(Y = y)} = P(V = x - y | Y = y)$$
This gives us:
$$ \int x P_{X | Y}(x | Y) dx = \int x \frac{P_{X,Y}(x,y)}{P_{Y}(y)} dx = \int x \frac{P_{V | Y}(V = x - y | Y = y) P_{Y}(y)}{P_{Y}(y)} dx = \int x P_{V | Y}(x - y | y) dx$$
If we know $Y = y$, this becomes simply $\int x P_{V}(x - y) dx$.
Edit: various notational corrections
