Mutual existence of sum and product integrals

Jon Craig Helton

Vol. 56 (1975), No. 2, 495–516

DOI: 10.2140/pjm.1975.56.495

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 ∫ _a^bG exists if and only if _x ∏ ^y(1 + G) exists for a ≦ x < y ≦ b. If each of lim_x→p + H(p,x), lim_x→p⁻H(x,p),lim_x,y→p⁺H(x,y) and lim_x,y→p⁻H(x,y) exists, G has bounded variation on [a,b] and either ∫ _a^bG exists or _x ∏ ^y(1 + G) exists for a ≦ x < y ≦ b, then ∫ _a^bHG and ∫ _a^bGH 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 ∫ _a^bG exists and ∫ _a^b|G −∫ G| = ν if and only if _x ∏ ^y(1 + G) exists for a ≦ x < y ≦ b and

Mathematical Subject Classification 2000

Primary: 46G10

Secondary: 46H99

Milestones

Received: 1 November 1973

Published: 1 February 1975