On the extension of interval functions

Abstract

Introduction. The problem of extending the range of definition of a function defined on a class of elementary figures-intervals, rectangles-has been treated in various ways in the literature. In the theory of Lebesgue measure a particular function-length of interval (area of rectangle)-is extended in a completely additive way to an additive class of sets. In the general extension problem we start, say, with a function (real, single-valued, and finite) of intervals ?(I) and extend the range of definition to an additive class of sets obtaining a function 4(E) which is completely additive and which has the property that 4(E) =O(I) whenever E is the set I. But what is the interval I? A priori 4,(I) is defined on a class of intervals I, where I is considered neither open nor closed but merely as an interval. From the viewpoint of 4(E) an interval I must be considered as a definite point set-a closed interval, an open interval, a semi-open interval, and so on. Corresponding to open intervals and to closed intervals, 4(E) gives rise to two interval functions: oi(I) = P(I'), ?2(I) = 4(I1) where I' is understood to be closed and 10 open. If +(I) =+(I) identically, then 4(E) is an extension of +(I) considered as a function of closed intervals; if 4(I) =02(I) identically, then 4?(E) is an extension of f(I) considered as a function of open intervals. As a starting point in the general extension problem, the function ?>(I) has been considered, somewhat artificially and arbitrarily perhaps, a function either of closed intervals or of open intervals (see, for example, [1O])(1). Extensions 4(E) which have the property that 4?(I') = 4(10) identically are of particular interest since then 4(E) is an extension of 4(I) whether I be considered open or closed. The main results of the paper concern the existence of B-extensions, a precise definition of which is given in ?1.6. Suffice it to say here that if 4(E) is a B-extension of 4(I) then 4?(I') =(I1) =0(I). The idea of a B-extension was suggested by a result of Burkill [2] which we shall review in ?1.5. Burkill's theorem on extension is stated in terms of a sufficient condition while our results on B-extensions are stated in terms of necessary and sufficient conditions. In Part 1 we explain notation, define terms, and summarize results. In Part 2 we present a proof of a theorem (Theorem 1) which states a necessary and sufficient condition that a non-negative function of closed intervals