  The main purpose of this paper is to give a vector lattice version of a
Theorem by Burkholder about convergence of martingales. The proof is based on a
vector lattice analogue of Austin's sample function theorem, proved recently by
Grobler, Labuschagne and Marraffa and on a new characterization of elements of
the sup-completion of a universally complete vector lattice which do not belong
to the space.
