On a convergence theorem for stochastic processes that have bounded sum of expected differences.

Question

Let $ \{ Z_t \}_{ t \in \mathbb{N} }$ be a real valued positive stochastic process defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P} )$ and adapted to a filtration $ \{ \mathcal{F}_t \}_{t \in \mathbb{N}}$. Let $$ \sum_{t \ge 1} E[ \boldsymbol{1}_{F_t}( Z_{t+1} - Z_t ) ] < \infty $$ where $\boldsymbol{1}_{F_t}$ represents the indicator function on the set $$F_t := \{ \omega \in \Omega | E[ Z_{t+1}(\omega) - Z_t(\omega) | \mathcal{F}_t ] > 0 \}$$ Then is it true that $$ Z_t \xrightarrow[]{\text{a.s.}} Z_{\infty} \ge 0 $$ where $Z_{\infty}$ is an integrable random variable.