Convergence of a typical martingale (A remark on the Doob theorem)

We study convergence behavior of discrete martingales with values in the interval [ 0 , 1 ] from a measure theoretical point of view as well as from a topological one. We show that almost all martingales converge to 0 or 1 almost everywhere. On the other hand, a typical martingale diverges on a comeager set.