Is $\sup_{n\in N} E[X_n] < \infty$?

Question

Here is the problem:

Let {$X_n$}$_{n\geq0}$ be a non-negative decreasing sequence of RV's, i.e. $X_n\leq X_{n-1}$, $\forall n \geq1$, such that $P(0\leq X_0<\infty)=1$.

Is it true that $\sup_{n\geq1}E(X_n)<\infty$ ? And if $E(X_0)<\infty?$

Well if it is true that $\sup E(X_n)\leq E(\sup X_n)$, then we are done since $\sup_{n\geq1}X_n=X_1$, and in that case we can say that $\sup_{n\geq1}E(X_n)<\infty$ if $E(X_0)<\infty$, by the monotone convergence theorem. So I would say no to the first question and yes to the second.

Am I right? is the inequality $\sup E(X_n)\leq E(\sup X_n)$ generally true for monotone sequences?

As always, many thanks for any help.

Answer 1

$EX_n$ is decreasing. $sup_{n\geq 1} EX_n=EX_1$ so $sup_n EX_n<\infty$ iff $EX_1 <\infty$. But $P(0\leq X_0 <\infty)=1$ does not imply that $EX_1 <\infty$ but if $EX_0 <\infty$ then $EX_1<\infty$ too.