Additive functionals on a space of continuous functions. I

Abstract

In a recent paper by the same title [II], hereafter referred to as Part I, R. H. Cameron and the present author investigated the structure of the class of additive functionals of class L2 on the Wiener space C [VI] and obtained certain other results concerning functionals additive and merely measurable on C. Recently Cameron conjectured that every functional additive and measurable on C was of class L2(C), and hence that the characterization and representation theory previously developed applied to all functionals additive and measurable on C. It is the purpose of this paper to establish Cameron's conjecture. Unfortunately the proof is not easy and depends on a number of preliminary results. However, the techniques of proof are novel enough to be of some interest. In addition, the results permit considerable strengthening of our earlier theorems and a curious characterization of the class of continuous linear functionals on the Hilbert space of real-valued functions of class L2 on the unit interval. We begin by introducing a number of preliminary concepts and theorems. DEFINITION 1. Let {I'pk} be a C. 0. N. set (complete orthonormal set in the L2 sense) on [0, 1], each of whose elements is of bounded variation, and let f(t)x E L2 on [0, 1]. Let fjt) be the nth partial sum of the orthogonal development of f(t) in terms of {I ik ; that is, let fn(t) = EL' Ckkp A(t), where Ck = If (t)k(t) dt. Then we define the P. W. Z. (Paley-Wiener-Zygmund) integral f f(t) dx(t) by the equation ff(t) dx(t) = lim ffn(t) dx(t), x E C O ~~~~~n -o 00 whenever the limit on the right exists. It has been shown by Paley, Wiener, and Zygmund [IV] that the limit defining the P. W. Z. integral exists for almost all x E C and that this limit is essentially independent of the particular choice of the C. 0. N. set {Ipk I in the sense that if IOk} and {lk* } are any two C. 0. N. sets all of whose elements are of B. V., then 1 1 (, f f(t) dx(t) = (O*) ] f(t) dx(t) for almost all x E C. It has also been o ~ ~~~~ I shown in [IV] that f fn(t) dx(t) converges in the L2(C) mean to a f(t) ax(t) that the P. W. Z. integral agrees with the ordinary Riemann-Stieltjes integral almost everywhere on C when f(t) is of B. V., and that if G(u1, **, un) is