# Difference between $E(y|X_1,X_2)$ and $E(y|\cup_{i=1,2}X_i)$

Suppose that $y$ is a random variable and $X_1$ and $X_2$ random vectors. Let $Z$ be the random vector that collects the distinct elements in $X_1$ and $X_2$. Can you please explain the difference (if any) between $$E(y|X_1,X_2)\quad\text{and}\quad E(y|Z)?$$ If you can, also please point me to an accessible reference so that I am better equipped to overcome this sort of confusion. I am familiar with the basics of measure theory and the measure-theoric definition of conditional expectations. Thank you.

• None.    – Did Nov 17 '14 at 14:55
• @Did Can you please elaborate, even by writing a short proof? I have always found the measure theoric definition hard to apply to practical situation like this. – yurnero Nov 17 '14 at 14:56
• Sure: the sigma-algebras $\sigma(X_1,X_2)$ and $\sigma(Z)$ coincide. – Did Nov 17 '14 at 14:58