A unifying Radon-Nikodym theorem for vector measures

Abstract

It is a known fact that certain derivation bases from martingales with a directed index set. On the other hand it is also true that the strong convergence of certain abstract martingales is a consequence of the Radon-Nikodym theory for vector measures (cf. Uhl, J. J., Jr., Trans. Amer. Math. Soc. 145 1969, 271–285). Many other connections and applications of the latter theory with multidimensional problems in stochastic processes and representation theory are known (cf. Dinculeanu, N., Studia Math. 25 1965, 181–205; Dinculeanu, N., and Foias, C., Canad. J. Math. 13 1961, 529–556; Rao, M. M., Ann. Mat. pura et applicata 76 1967, 107–132; Rybakov, V. I., Izv. Vysš. Učebn. Zaved. Matematika 19 1968, 92–101; Rybakov, V. I., Dokl. Akad. Nauk SSSR 180 1968, 620–623). Starting from various vantage points, many authors have proposed several hypotheses for establishing abstract Radon-Nikodym theorems. In view of the great interest and importance of this problem in the areas mentioned above, it is natural to obtain a unifying result with a general enough hypothesis to deduce the various forms of the Radon-Nikodym theorem for vector measures. This should illuminate the Radon-Nikodym theory for vector measures and stimulate further work in abstract martingale problems. In this paper the first problem is attacked, leaving the martingale part and other applications for another study. The main result (Theorem 7 of Section 2) provides the desired unification and from if the Dunford-Pettis theorem, the Phillips theorem and several others are obtained. As martingale-type arguments are constantly present, a careful reader may note the easy translation of the hypothesis to the martingale convergence problem but we treat only the Radon-Nikodym problem using the language of measure theory and linear analysis.