If $E(X_{n+1}\vert \mathcal{F}_{n})\leq X_{n}+Y_{n}$ and $\sum Y_{n} < \infty$ how can I define a supermartingale?
Prove a.s. convergence of $(X_n)_n$ satisfying $E(X_{n+1} \mid F_n) \leq X_n+Y_n$ for $\sum_n Y_n<\infty$