Limit of random variables with martingale

Question

Let $\{(X_n,\cal F_n)\}$ be a supermartingale, and suppose $X_n$ converges almost surely to some $X_{\infty}\in L^1$. Are the following true:

1. For a fixed $n$, Is it true that $\lim_{m\to \infty}E(X_m|\cal F_n)$=$E(X_{\infty}|\cal F_n)$? Or more generally, is it true that for any subfield $\cal G$, without the supermartingale condition, we still have $\lim_{m\to \infty}E(X_m|\cal G)$=$E(X_{\infty}|\cal G)$?

2. Assume the supermartingale condition. We know that $E(X_m|\mathcal F_n)\leq X_n$ for all $m>n$. But is it true that $E(X_{\infty}|\mathcal F_n)\leq X_n$?

3. Is it true that $\lim_n E(X_{\infty}|\mathcal F_n)=X_{\infty}$? The idea is that as $n$ becomes larger, we would like to have $X_{\infty}$ be $\mathcal F_{\infty}$ measurable, where $\mathcal F_{\infty}=\bigcup_{n=1}^{\infty}\mathcal F_n$. Again, this property seems to be independent of the supermartingale condtion.

Edited: My idea is as the following: We are proving the following theorem: Let $\alpha\leq \beta$ be two stopping times, and $\{(X_n,\cal F_n)\}$ is a supermartingale. Suppose $X_n$ converges almost surely to some $X_{\infty}\in L^1$. then $\{(X_\alpha,\mathcal F_\alpha),(X_\beta,\mathcal F_\beta)\}$ is a two term martingale.

We first proved the case when $X_{n}\geq 0$ and $X_{\infty}=0$. Next for the general case, we set $X_n'=E(X_{\infty}|\mathcal F_n)$, and $X_n''=X_n-X_n'$. We hope to apply the special case to $X_n''$ to get the answer. Now here is where I got stuck.

I include the source of the problem which is from a lecture in our school. http://www.math.cuhk.edu.hk/course_builder/1516/math6081b/2016-Stoch5.pdf The theorem is proved in Page 36.

Answer 1

The answer to most of your questions is no.

The first is not true since you can have a situation in which the conditional expectations of the supermartingale property are strict. So you will never get equality. Also you need L1 convergence for expectations to converge. Maybe you can polish this statement up a bit to ensure this (hint: take a supermartingale with constant expectation which is uniformly integrable). Also look at: http://www.math.leidenuniv.nl/~spieksma/colleges/sp-master/sp-hvz1.pdf, theorem 2.2.13 en 2.2.14.

The second is also not true by theorem 2.2.14 in this reader: http://www.math.leidenuniv.nl/~spieksma/colleges/sp-master/sp-hvz1.pdf. The condition which you need is formulated in the theorem.

The third is true by, in the above reader, Levy's upward theorem (2.2.15).