Vol. 56, No. 2, 1975

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Mutual existence of sum and product integrals

Jon Craig Helton

Vol. 56 (1975), No. 2, 495–516
Abstract

Functions are from R×R to N, where R denotes the set of real numbers and N denotes a normed complete ring. If G has bounded variation on [a,b], then abG exists if and only if x y(1 + G) exists for a x < y b. If each of limxp + H(p,x), limxpH(x,p),limx,yp+H(x,y) and limx,ypH(x,y) exists, G has bounded variation on [a,b] and either abG exists or x y(1 + G) exists for a x < y b, then abHG and abGH exist and x y(1 + HG) and x y(1 + GH) exist for a x < y b. If G has bounded variation on [a,b] and ν is a nonnegative number, then abG exists and ab|G G| = ν if and only if x y(1 + G) exists for a x < y b and

∫ b
a |1+ G − Π(1+ G )| = ν.

Mathematical Subject Classification 2000
Primary: 46G10
Secondary: 46H99
Milestones
Received: 1 November 1973
Published: 1 February 1975
Authors
Jon Craig Helton