Can anyone help me in an intuitive manner how to simplify the following conditional probabilities and expectations - $E(Y|X)|X$ and $E(E(Y|X)|X)$ -? I am trying to use their literal interpretation (the expectation of $Y$ conditional on $X$ conditinalized on $X$) to see how to move forward, but that has failed me so far. any tips welcome.
First, the entity $E(Y|X)|X$ does not exist (and the phrase "the expectation of $Y$ conditional on $X$ conditinalized on $X$" is pure mumbo-jumbo).
Second, $E(E(Y|X)|X) = E(Y|X)$ by definition, because the random variable $E(Y|X)$ is $σ(X)$-measurable and because $E(Z|X) = Z$ for every $σ(X)$-measurable (integrable) random variable $Z$.