1. expectation

suppose you have a sequence of random variables $(X_n)_{n \geq 0}$. Then what does $sup_n E[X_n]$ mean? it does not make sense to me since if n is fixed then $E[X_n]$ is a constant. Is it the same thing as $lim sup_{n-> \infty} E[X_n]$?

2. Originally Posted by Kat-M
suppose you have a sequence of random variables $(X_n)_{n \geq 0}$. Then what does $sup_n E[X_n]$ mean? it does not make sense to me since if n is fixed then $E[X_n]$ is a constant. Is it the same thing as $lim sup_{n-> \infty} E[X_n]$?

$E[X_n]$ is a constant
So all you're asking is what is a supremum of a sequence $a_n$
limsup is the limit of the largest subsequence.

3. expectation

so $sup_n E[X_n]=sup_{j \leq n }E[X_j]$?