Conditional expectation Doubt

Question

Let $\{X_n\}$ be non-negative integrable random variables on $(\Omega, \mathscr{F}, \mathscr{P})$ adpated to a filtration $\{ \mathscr{F_n}\}$ and bounded a.s. by a constant $C < \infty$. Let $\{ \mathscr{F_{-1}}\}=\{\phi, \Omega\}$. Show that the sets $\{\sum_{n=0}^{\infty} X_n < \infty \}$ and $\{\sum_{n=0}^{\infty} E[X_n |\mathscr{F_{n-1}]} < \infty \}$ coincide a.s.

How to proceed here. Hint Enough.

Answer 1

The key point is the following theorem (3.1 p.239 in Durrett's book Probability: Theory and Examples, second edition):

Let $(S_j)_{j=1}^{+\infty}$ be a martingale, where $S_j=\sum_{k=0}^jX_k$ and $|X_k(\omega)|\leqslant M$ almost surely ($M$ is constant). Define $$C:=\{\omega,\lim_{n\to\infty}S_n(\omega)\mbox{ exists and is finite}\} $$ $$D:=\{\omega, \limsup_{n\to \infty}S_n(\omega)=+\infty,\liminf_{n\to \infty}S_n(\omega)=-\infty.$$ Then $\mathbb P(C\cup D)=1$.

The proof goes as follow: assume that $X_0=0$ and fix $K$ and define $N:=\inf\{n,X_n\leqslant -K\}$. Notice that $(S_{n\wedge N}+K+M)_{n\geqslant 1}$ is a positive submartingale, hence $\lim_{n\to \infty}S_n$ exists on $\{N=\infty\}$. Letting $K\to +\infty$, we can see that the limit exists provided that $\liminf_n S_n>-\infty$.