Strong convergence of positive subpramarts in Banach lattices

Abstract

The author gives a (strong) sufficient condition that insures the norm convergence of positive subpramarts valued in Banach lattices with the Radon-Nikodym property. The method consists of extending a lemma of Neveu about sequences of real-valued submartingales to be able to apply the techniques of Davis-Ghoussoub-Lindenstrauss. A recent result of M. Talagrand [Isr. J. Math. 44, 213-220 (1983; Zbl 0523.46016)] gives the result without the additional condition imposed by the author. This theorem of Talagrand states that every separable Banach lattice with the Radon-Nikodym property is actually a dual Banach lattice.