J. A. Chatfield

Copyright © 1978 J. A. Chatfield. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Suppose N is a Banach space of norm |•| and R is the set of real numbers. All integrals used are of the subdivision-refinement type. The main theorem [Theorem 3] gives a representation of TH where H is a function from R×R to N such that H(p+,p+), H(p,p+), H(p−,p−), and H(p−,p) each exist for each p and T is a bounded linear operator on the space of all such functions H. In particular we show that TH=(I)∫abfHdα+∑i=1∞[H(xi−1,xi−1+)−H(xi−1+,xi−1+)]β(xi−1)+∑i=1∞[H(xi−,xi)−H(xi−,xi−)]Θ(xi−1,xi)where each of α, β, and Θ depend only on T, α is of bounded variation, β and Θ are 0 except at a countable number of points, fH is a function from R to N depending on H and {xi}i=1∞ denotes the points P in [a,b]. for which [H(p,p+)−H(p+,p+)]≠0 or [H(p−,p)−H(p−,p−)]≠0. We also define an interior interval function integral and give a relationship between it and the standard interval function integral.