Given two real valued random variables $X$ and $Y$ on $(\Omega, \Sigma)$, we can define a random variable $Z=(X,Y)$ on $(\Omega \times \Omega, \Sigma \otimes \Sigma)$. But what is the expected value of $Z$ then?
I mean we can still use the "normal" definition of the expected value, but the expected value would be in $\mathbb R^2$, not in $\mathbb R$. Is that okay? And if so, how do terms like "uncorrelated" translate? Do we want $E(Z_1 Z_2)-E(Z_1)E(Z_2)= (0,0)$ then?