Folia Mathematica Acta Universitatis Lodziensis Vol. 12, No. 1, pp. 25–37 c  2005 for University of  L´ od´ z Press STOCHASTIC PROCESSES OF VECTOR VALUED PETTIS AND MCSHANE INTEGRABLE FUNCTIONS VALERIA MARRAFFA ‡ Abstract. Convergence theorems of Pettis integrable martingales and more general stochastic processes taking values in a Banach space without the weak Radon–Nikodym property, are considered. Also McShane integrable martin- gales are studied. 1. Introduction In this paper we study properties of stochastic processes, consisting of weakly measurable functions, taking values in a Banach space. For a Banach– valued martingale of Pettis integrable functions, the weak Radon–Nikodym property is equivalent to the convergence in Pettis norm (see [8]). Without assuming this property we ask for which smaller class T of functionals f the a.s. convergence of fX n to fX for f ∈ T implies the convergence of X n to X in Pettis norm. For Bochner integrable stochastic processes it was shown that T can be a total set (see [2] and [7]). In Section 3 we prove analogous results for Pettis – integrable martingales. In [6] convergence theorems for more general sequences are proved under the hypothesis that the range is relatively weakly compact or has the weak Radon-Nikodym property. Using decomposition theorems we obtain convergence results for stochastic processes taking values in a space which does not possess the weak Radon–Nikodym property (see Theorem 4 and Theorem 5). In the fourth section martingales of McShane integrable functions are considered. In particular it is proved that a closed martingale converges in the McShane seminorm (Proposition 1). For a Pettis integrable closed martingale the same is true if the range of the indefinite integral is norm relatively compact (see [8] Corollary 1). Also a convergence theorem for McShane integrable martingales is proved (see Theorem 7). Supported by MURST of Italy. ‡ University of Palermo, Department of Mathematics, Archirafi 34 St., 90123 Palermo, Italy. E-mail: marraffa@math.unipa.it. Key words and phrases: Pettis integral, martingale, McShane integral, Banach space valued processes. AMS subject classifications: 60G48, 28B05.