What can I say about expected value of expected value of x given y: $E[E[X|Y]]$

Question

My question is about Y that is discrete and for some random variable X, but if its having a meaning for Y that is continuous then please give that case your attention.

What can I say about $E[E[X|Y]]$?

I know that E[X|Y] is random variable, so It's not trivial case when we calculate expected value of just a number.

And what about $E[E[X|Y]|Y]$? does something like this have a meaning? if it's, then does for some function $g$ (for simplicity, assuming g with suitable range and continuous) it's true to say that: $$E[g(Y)E[X|Y]|Y]=g(Y)E[E[X|Y]|Y]$$ Because of a theorem that I seen: $$E[g(Y)X|Y]=g(Y)E[X|Y]$$

Answer 1

If $X$ is measurable wrt $\sigma(Y)$ i.e. the $\sigma$-algebra generated by random variable $Y$ then $\mathbb E[X\mid Y]=X$.

This can be applied on $Z:=\mathbb E[X\mid Y]$ because $\mathbb E[X\mid Y]$ is by definition measurable wrt $\sigma(Y)$.

This results in $\mathbb E[Z\mid Y]=Z$ or equivalently: $$\mathbb E[\mathbb E[X\mid Y]\mid Y]=\mathbb E[X\mid Y]$$

Answer 2

It is a basic fact about conditional expectations that $E(E(X|Y))=EX$.