On Stochastic Set Functions. Iii. I. Preliminary Remarks on Totals

In the present paper we are concerned with two topics: the probability distributions of the random variables consisting of a completely additive s t o c hastic set function and the discontinuities of the realizations by supposing the latters to be completely additive number-valued set functions. Some theorems of Chapter III are analogous to those well-known in the theory of stochastic processes with independent increments. (cf. mainly 13]). There are questions arising in a natural way in the eld discussed in this paper which we did not consider in detail such as the representation of a stochastic set function as a sum of an atomless and an atomic part, etc. These problems are not very diicult to solve with well-known methods or with our results. We refer to the deenitions and notations of the previous papers 19], 20] which a r e used here without any remark. Some new deenitions and notations will also be introduced at the place where they are needed. The notion of the \total", which is a generalization of the Burkill integral, was introduced by M. Cotlar and Y. Frenkel 4]. A special case of this was generalized by t h e author 22] for functions with values in a Banach algebra. These notions and the theorems proved in connection with them nd an extensive application in the present paper. In this Chapter we f o r m ulate some deenitions and theorems relative to the totals in a form convenient for our treatment. We do not strive in the deenitions for the most general versions and mention only the simplest theorems. To other theorems proved in the paper 22] w e refer at the place of their application. Let H be an abstract set and K a semi-ring of sets consisting of some subsets of H. This means the following: Definition 1. 1 A class of sets K is called a semi-ring if for every pair of sets A 2 K , B 2 K for which A B there is a nite system of sets C k 2 K (k = 1 : : : r) s u c h that B ; A = r X k=1 C k 1 This notion of semi-rings is due to A. Cs asz ar.