I am working on the following problem: Let $(X_n),n\geq1$ be a non-decreasing sequence of non-negative random variables. Assume that $\sup_{n\geq1}E(X_n)<\infty$. Prove the following:
a) $(X_n)_{n\geq1}$ converges almost surely
b) Denote the limit with $X$. Show that $E(X)<\infty$
c) $E(X|\mathcal{G})=\lim_{n\to\infty}E(X_n|\mathcal{G})$
I think b) and c) can be proven trivially with Fatou's lemma, e.g.:
$E(X)=E(\liminf X_n) \leq \liminf E(X_n)< \infty$. Does that also imply the existence of $X$? I think it doesn't, but I can't find a theorem that would guarantee the existence. Any ideas for a)?
$(X_n)$ is non-negative and non-decreasing. This implies that $X=\lim X_n$ exists (but it may take the value $\infty$). Fatou's Lemma gives $EX <\infty$ and this implies that $X$ is almost surely finite valued.
Now $Z=\lim_n E(X_n|\mathcal G)$ exists for the same reason. To prove that $Z=E(X|\mathcal G)$ note that $Z$ is $\mathcal G-$ measurable and prove that $\int_A Z dP=\int_A XdP$ for all $A \in \mathcal G$. For this apply Monotone Convergence Theorem.