Martingales in a 𝜎-finite measure space indexed by directed sets

Abstract

Introduction. Martingale theory for a linearly ordered index set in a finite measure space has been systematically discussed by Doob [4]. As convergence theory and system theory are the two main topics in martingale theory, Helms [6] has extended Doob's results on mean convergence to martingales in a finite measure space, indexed by directed sets, and Krickeberg [8; 9] extended the results of pointwise convergence and stochastic convergence to martingales in a a-finite measure space, indexed by directed sets. Krickeberg uses measure algebra and lattice theoretic methods, and in [8] makes the assumption in pointwise convergence that the martingales have some covering property, which is called Vo in his paper. Dieudonne [3] has given a martingale which is not pointwise convergent and has non-negative, bounded functions in a finite measure space, indexed by a countable directed set. Therefore, some additional conditions like Krickeberg's Vo to the usual ones seem to be necessary for pointwise convergence. However, the condition Vo is too weak to be the only additional condition needed for pointwise convergence of martingales indexed by noncountable directed sets. Some conditions such as terminal separability are needed. For the system theory, Bochner [2] has given some extensions to martingales indexed by directed sets, but without proofs. Some of his results are, unfortunately, not true. Snell [14] has defined regularity (which is closely related to system theorems) of martingales in a finite measure space, indexed by positive integers, and has given some sufficient conditions. He has not discussed the necessary conditions, nor the relation between regularity and the system theorems. In this paper, ?1 gives preliminary definitions and notation. ?2 is devoted to the definition of conditional expectation in a a-finite measure space W, and ?3 to that of martingales indexed by directed sets in W as well as to individual system theorems and inequalities. In ?4, a general pointwise convergence theorem is proved, and a new kind of convergence theorem, based on a differentiation theorem of interval functions [13, p. 192] is given. We