Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Additivity and indefinite integration for McShane's $ P$-integral

Author: C. H. Scanlon
Journal: Proc. Amer. Math. Soc. 39 (1973), 129-134
MSC: Primary 26A42
MathSciNet review: 0315060
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Abstract: Suppose $ [a,b]$ is a closed real number interval, $ A = \{ (p,q];(p,q] \subset (a,b]\} $, and $ U$ is a real valued function on $ [a,b] \times A$. For $ c \in (a,b)$, necessary and sufficient conditions are given for $ P$-integrability of $ U$ on $ (a,b]$ and $ (c,b]$ to imply $ P$-integrability on $ (a,b]$. Suppose $ U$ is $ P$-integrable on $ (a,b]$ and $ g(x) = P\smallint _\alpha ^xU$ for each $ x \in (a,b]$. Necessary and sufficient conditions are given for $ g$ to be respectively continuous, bounded, and of bounded variation.

References [Enhancements On Off] (What's this?)

  • [1] E. J. McShane, A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals, Memoirs of the American Mathematical Society, No. 88, American Mathematical Society, Providence, R.I., 1969. MR 0265527
  • [2] A. Kolmogoroff, Unterschungen über den Integralbegriff, Math. Ann. 103 (1930), 654-696.
  • [3] Ralph Henstock, Theory of integration, Butterworths, London, 1963. MR 0158047

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Keywords: Integration, indefinite integration, additivity
Article copyright: © Copyright 1973 American Mathematical Society