On saltus-functions and interval-surfaces.

As regards terminology and notation we shall follow the Theory of the Integral by Saks as a rule (see the list of references at the end). The mentioned treatise will be quoted very frequently in the sequel and we shall henceforward refer to it simply as Saks for short. The space that will be basic for all our considerations is a fixed Euclidean space Rm of dimension m•†1. When we specialize our space (to be, say, the straight line R1), we shall say so explicitly. Thus the various sets that will be considered in the sequel are subsets of Rm, unless another meaning is obvious from the context. We shall always understand by intervals, by themselves, closed nondegenerate intervals in Rm. It may be remarked that, contrary to the use in Saks, we thus do not regard the void set as an interval. Further more, the letters ƒÂ and ƒÃ will invariably stand for finite positive numbers. We shall be chiefly concerned with additive interval-functions of bounded variation, defined for the intervals in Rm, and with additive set-functions, de fined for the bounded Borel sets in Rm. In what follows we shall refer to them for brevity of wording simply as additive interval functions and additive set functions, even when we use such expressions as an arbitrary additive interval function, and so on. The present paper is divided into two chapters. In the first chapter we aim at obtaining a complete extension to Rm of the theory of saltus-functions developed in the one-dimensional case in Chapter III of Saks. The reasonings that are valid for Rl are no more valid for higher dimensions. We are thus obliged to define the saltus-functions in a different manner than in Saks, and we shall deduce their fundamental properties by way of diverse arguments of auxiliary character. As a typical result of this chapter we may mention that of • ̃40 to the effect that if F is an additive interval-function and W is its ab solute variation, then W(F*;X)=W*(X) for every bounded Borel set X which is disjoint with the hyperplanes of discontinuity of F. This constitutes a natural extension to m dimensions of the theorem given in Saks on p. 99. A few words of caution are necessary for our use, throughout the paper, of the notation F* appearing above. The function F* is defined on p. 64 of the Saks treatise for all sets of the space Rm. It is mostly convenient for our purpose, however, to consider this set-function only for the bounded Borel sets in Rm. To prevent ambiguity, we therefore agree to make the convention that the symbol F* should mean, in our sense, the restriction of F* in the Saks sense to the class of all bounded Borel sets and that, when there is need, a special notation F* should be used to denote F* in the Saks sense. Thus F* is defined for all the sets in Rm, and is an outer Caratheodory measure if F is