article · 2015
ADDENDUM TO: HELLINGER INTEGRALS AND SET FUNCTION DERIVATIVES
Wayne C. Bell
Cited by 0Abstract and details only
Abstract
ABSTRACT. Using methods from the paper in the title above we prove that a Radon-Nikodym type theorem obtained by Hatta and Wayment does, as the authors conjectured, generalize to the finitely additive case. In Theorem 3.1 and 4.1 of [1] Hatta and Wayment proved Theorem 7 below assuming the additional hypothesis that F is a a-field and •u (hence each of 7 and v) is countably additive. The extended v-integral used there is the same (see [2]) as the integral considered here. Their proof involved the Lebesgue integral and they conjectured that an analogous argument could be carried out in the Emitely additive case. What follows is a substantially different approach. We will prove a variation of one of the corollaries of Kolmogoroff's differential equivalence theorem and obtain Theorem 7 as a consequence of this and certain results of W. D. L. Appling which can be found in [3]. Suppose S is a set, F is a field of subsets of S, r(F) the set of real valued functions on F, b(F) the set of bounded elements of r(F) and ba(F) the set of additive (on disjoint sets) elements of b(F). For G _C r(F) denote by G + the set of nonnegatively valued elements ofG. Let # be in ba(F) +. The integral considered below is the refinement limit of sums over finite subdivisions of S by elements of F. Ifo • C r(F), then fso•(E) exists iff fAO•(E) exists for each A C F, in which case we write the indefinite integral, fo•, by which we shall mean the function whose value ateach EG F is fEo•(I). For further details concerning the integral, undeEmed terminology and the use of the function c as a substitute for the Hahn decomposition theorem we refer the reader to [3]. As in [3] we adopt the convention that a/0 = 0 for a GR. We will need the following small variation of a theorem of W. D. L. Appling (see
Cite this paper
Bell, W. C. (2015). ADDENDUM TO: HELLINGER INTEGRALS AND SET FUNCTION DERIVATIVES.
Bell, Wayne C. “ADDENDUM TO: HELLINGER INTEGRALS AND SET FUNCTION DERIVATIVES.” 2015.
Bell, Wayne C. 2015. “ADDENDUM TO: HELLINGER INTEGRALS AND SET FUNCTION DERIVATIVES.”
Bell, W.C. (2015) ‘ADDENDUM TO: HELLINGER INTEGRALS AND SET FUNCTION DERIVATIVES’.
@article{bell2015addendum,
title = {{ADDENDUM TO: HELLINGER INTEGRALS AND SET FUNCTION DERIVATIVES}},
author = {Bell, Wayne C.},
year = {2015}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.