As I keep reading probability books, there are always some issues that no one considers.
for $\omega \in \Omega$ and $X$, $Y$ independent random variable we define $Z(\omega )=X(\omega )\cdot Y(\omega)$, So if $E[X]$ , $E[Y]$ , $E[Z]$ defined, we know that $E[X]\cdot E[Y]=E[Z]$.
But, I really curious whether there's a situation when $E[X]$, $E[Y]$ defined, but $E[X\cdot Y]$ ($E[Z]$) is $\infty$ or even Diverging? I wasnt able to think of an answer.
(Is it ok to post more than one question in the same day?)