I had this basic conjecture about random variables. Let $X$, $Y$ random variables on the same measure space.
Original claim: If for elementary events $\omega_1,\omega_2$ we have $Y(\omega_1)=Y(\omega_2)$, then also $E[X|Y](\omega_1)=E[X|Y](\omega_2)$.
I think the claim as it is cannot be correct since the expectation value is a random variable, defined therefore up only to equivalence, and we can change representative changing the value on a zero measure set. Nevertheless I think intuitively there should be a related statement because what the expectation value $E[X|Y]$ is doing, to my understanding, is taking a local mean of $X$ over a set where $Y$ assumes the same value. My questions are:
- does it exist a version of this claim that makes sense from a measure-theoretic point of view ? If this version of the claim exists, this new version is also true?
The conjecture is basically right. The conjecture, phrased differently, is that $E[X|Y]$ is a function of $Y$. A version of the conditional expectation (by definition) is measurable wrt the conditioning variable, thus a function of the conditioning variable.