ON CONVERGENCE OF -BOUNDED MARTINGALES INDEXED BY
DIRECTED SETS
Annie Millet
Louis Sucheston
Abstract: Let be an increasing family of -algebras indexed by a directed set In
this paper it is shown that every -bounded real-valued martingale converges essentially if
and only if a weak type of maximal inequality holds for all martingales. A new covering
condition C stated in terms of multivalued stopping times is introduced and characterized in
terms of maximal inequalities. C is shown to be strictly weaker than the Vitali condition V,
than SV (see [15]), and also sigma-SV. Under C, -bounded martingales taking
values in a Banach space with the Radon-Nikodým property converge essentially.
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -