Nonnegative martingale ($M_n\ge 0$)

A nonnegative martingale takes nonnegative values almost surely at each time. In discrete time its expectations are constant, so it is bounded in $L^1$ and the Martingale convergence theorem gives a finite integrable almost sure limit. Its expectations need not converge to the expectation of that limit; uniform integrability is needed for convergence in L1.