Conditional Random Variable and Iterated Expectation First, I can't seem to find a definition of a "conditional random variable"; I understand the concepts of conditional expectation and conditional probability, but is there a notion for the random variable defined by $X|A$ for $A$ an event? Or more generally, is there a notion for the r.v. $X|Y$ for r.v.s $X$ and $Y$? 
I ask because I argued in a problem that a desired expectation is $\mathbb{E}( \mathbb{E}( X | A) | B ) $ for $A$ and $B$ two events, but I didn't actually compute $\mathbb{E}( \mathbb{E}( X | A) | B ) $  mathematically per say, but argued that it was equal to $\mathbb{E}( X | A \cap B)$ based on the logical construct of the problem and the idea of total expectation. 
My second question is, is it generally true that $\mathbb{E}( \mathbb{E}( X | A) | B ) = \mathbb{E}( X | A \cap B)$ for $A,B$ events? 
 A: One usually doesn't define "conditional random variables", but you could say: given random variable $X$ and event $A$ with $P(A) > 0$, consider the new sample space $A$ with probability
measure $P'$ given by $P'(B) = P(B|A)$ for $B \subset A$, and the random variable $Y$ on this sample space
such that $P(Y \ge y) = P(X \ge y|A)$.
Since ${\mathbb E}(X|A)$ is a number (if it is defined at all) rather than a random variable,
your ${\mathbb E}({\mathbb E}(X|A)|B)$ can only be ${\mathbb E}(X|A)$.
A: This is an old question, but I should point out that $E(X|A)$ can be interpreted as $E(X|\sigma(A))$. Here, as $A$ is just an event, $\sigma(A)=\{\emptyset, \Omega, A, A^c\}$. Also, seen this way, $E(X|A)$ need not be just a real number; it'll be a random variable which is constant on $A$ and $A^c$. A cheap counterexample is to take $X=1_A$, in which case $E(X|A) = 1_A$. 
A simple calculation shows that, per this interpretation, $$E[E(1_A|A)|B] = P(A|B)1_B+P(A|B^c)1_{B^c}$$ and $$ E(1_A|A\cap B) = 1_{A\cap B} + \frac{P(A\cap B^c)}{P((A\cap B)^c)}1_{(A\cap B)^c}.$$  
