I have just read a measure theoretical discussion of conditional expectations and conditional densities. I am confused about the following:
We know that for random variables $X$ and $Y$ (real valued, integrable, defined on $(\Omega,\mathcal A, P)$) $$\mathbb E(X \mid Y) = g(Y)=\mathbb E(X\mid Y=\cdot) \circ Y$$ ($P$-a.s.).
We also know that $$\mathbb E(X\mid Y=y)=\int x \frac{f(x,y)}{f_Y(y)}\ \mathsf d \lambda(x)$$ ($P_Y$ a.s.) and we call $$f(x\mid y):=\frac{f(x,y)}{f_Y(y)}$$ conditional density.
If $f(x\mid y)$ is a density, then it defines a probability measure $$\nu(A)=\int_{A}f(x\mid y)\ \mathsf d\lambda(x)$$ for all $A \in \mathcal B^1$, moreover this measure has a distribution function $$G(x) = \int_{(-\infty,x]}f(x\mid y)\ \mathsf d\lambda(x)=\frac{\int_{(-\infty,x]}f(x,y)\ \mathsf d\lambda(x)}{f_Y(y)}.$$
- Is there any insightful relationship between $\nu$ and $X,Y, \mathbb E(X\mid Y)$?
- Conditional CDF is defined as $F(x\mid y)=\frac{F_{X,Y}(x,y)}{F_Y(y)}$. Is there any insightful relationship between $G(x)$ and $F(x\mid y)$? If there is none, is not it strange that $F$ and not $G$ is called conditional CDF?