What is $E[E[Y|X]|X]$? What is $E[E[Y|X]|X]$ ?
Can I use the law of total expectation and say that it's equal to $E[Y|X]$ ?
If it's correct, could someone explain why?
 A: Let $\mathscr F$ be the $\sigma$-algebra generated by $X.$ Then
$E[Y|X]=E[Y|\mathscr F]$ is an $\mathscr F$-measurable random variable and we know that if $Z$ is a $\mathscr F$-measurable random variable, we have $E[Z|\mathscr F]=Z.$
A: I do not think this is the low of total expectation, at least not in its standard form.
Recall instead that:

*

*$E[A|B]=A$ if $A$ is $B-$measurable.

*$E[Y|X]$ is $X-$measurable.

So now put in the first property $A=E[Y|X]$ and $B=X$, that is your result.
A: The law of total expectation would $E[Y]=E[E[Y|X]]$. In this expression $E[Y|X]$ is a random variable that is a function of $X.$ $E[Y|X]$ changes with $X,$ assuming different realizations depending on different outcomes of $X=x,$ and by the law of total expectation, its expected value, across the support of $X$ is just $E[Y].$ Conditioning on $X$ the expression $E[Y|X]|X$ is redundant $E[Y|X]|X=E[Y|X],$ and in this sense it amounts to applying the same law of total expectation $E[E[Y|X]]=E[E[Y|X]|X]=E[Y].$ $\;$
