What's $\mathbb E[\mathbb E[Y|X]|Y]$? Is it well-defined?

Question

Consider two real random variables $X,Y$ and the conditional expectation $\mathbb E[Y|X]$, also a random variable. What is the conditional expectation $\mathbb E[\mathbb E[Y|X]|Y]$? Is it $=Y$? Is it $=\mathbb E[Y|X]$? Is it even well-defined?

Answer 1

Yes it is well defined, but it is not equal to anything 'obvious', here is an example.

Let $(X,Y)$ be jointly gaussian, each with mean 0, variance 1, correlation between $X$ and $Y$ is $\rho$, then

$E[Y|X] = \rho X$

$E[E[Y|X]|Y]=E[\rho X|Y]= \rho^2 Y$

NOTE: $E[E[Y|X]|Y]$ is $\sigma(Y)$-measurable random variable and $E[Y|X]$ is a $\sigma(X)$-measurable function. There is no reason why they are equal. the only obviou case when they are the same is either $Y$ is $X$–mesurable or $X$ and $Y$ are independent