Abstrakt
We investigate convergence of martingales adapted to a given filtration of finite alpha-algebras. To any such filtration, we associate a canonical metrizable compact space K such that martingales adapted to the filtration can be canonically represented on K.
We further show that (except for trivial cases) typical martingale diverges at a comeager subset of K. 'Typical martingale' means a martingale from a comeager set in any of the standard spaces of martingales. In particular, we show that a typical L-1-bounded martingale of norm at most one converges almost surely to zero and has maximal possible oscillation on a comeager set.