Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 (42 #436)
  • [2] A. Kolmogoroff, Unterschungen über den Integralbegriff, Math. Ann. 103 (1930), 654-696.
  • [3] Ralph Henstock, Theory of integration, Butterworths, London, 1963. MR 0158047 (28 #1274)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A42

Retrieve articles in all journals with MSC: 26A42


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0315060-X
PII: S 0002-9939(1973)0315060-X
Keywords: Integration, indefinite integration, additivity
Article copyright: © Copyright 1973 American Mathematical Society