Abstract
Lebesgue type integrals of mappings f: E → X, where X is a semiordered ring and E ⊂R, over a measure with values lying in X are studied. Usual properties of the integral are proved, including theorems on limits under the integral as well as those on absolute continuity and complete additivity of the (u, v)-integral. A theorem on the existence of a solution of a differential equation with (u, v)-derivatives is proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
B. Z. Vulikh, Introduction to the Theory of Semiordered Spaces [in Russian], Fizmatgiz, Moscow (1961).
V. I. Sobolev, “On a semiordered measure of sets, measurable functions and some abstract integrals, “Dokl. Akad. Nauk SSSR,91, No. 1, 23–26 (1953).
V. I. Sobolev and V. M. Shcherbin, “On differentiation of mappings of ℋ-spaces,” ibid.,225, No. 5, 1020–1022 (1975).
V. M. Shcherbin, “On Riemann integration of mappings of semiordered rings,” Dep. in VINITI, No. 2839–85, Moscow (1985).
V. M. Shcherbin, “Relatively uniform derivatives of mappings of ℋ-spaces and some theorems on local inversion of mappings,” Cand. Phys. Math. Sci. Dissertation, Tashkent (1979).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 547–555, July–August, 1992.
Rights and permissions
About this article
Cite this article
Sobolev, V.I., Shcherbin, V.M. Integration of mappings of semiordered rings. Ukr Math J 44, 488–495 (1992). https://doi.org/10.1007/BF01064883
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01064883