A note on the generalized Riemann integral

Pfeffer, Washek F.

doi:10.1090/S0002-9939-1988-0955000-4

A note on the generalized Riemann integral
HTML articles powered by AMS MathViewer

by Washek F. Pfeffer

Proc. Amer. Math. Soc. 103 (1988), 1161-1166

DOI: https://doi.org/10.1090/S0002-9939-1988-0955000-4

PDF | Request permission

Abstract:

We show that the generalized Riemann integral can be defined by means of gage functions which are upper semicontinuous when restricted to a suitable subset whose complement has measure zero.

References

S. I. Ahmed and W. F. Pfeffer, A Riemann integral in a locally compact Hausdorff space, J. Austral. Math. Soc. Ser. A 41 (1986), no. 1, 115–137. MR 846776
James Foran and Sandra Meinershagen, Some answers to a question of P. Bullen, Real Anal. Exchange 13 (1987/88), no. 1, 265–277. MR 923731
Ralph Henstock, Theory of integration, Butterworths, London, 1963. MR 0158047
Jiří Jarník, Jaroslav Kurzweil, and Štefan Schwabik, On Mawhin’s approach to multiple nonabsolutely convergent integral, Časopis Pěst. Mat. 108 (1983), no. 4, 356–380 (English, with Russian summary). MR 727536
Jaroslav Kurzweil, Nichtabsolut konvergente Integrale, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 26, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980 (German). With English, French and Russian summaries. MR 597703
Jean Mawhin, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, Czechoslovak Math. J. 31(106) (1981), no. 4, 614–632. MR 631606
Robert M. McLeod, The generalized Riemann integral, Carus Mathematical Monographs, vol. 20, Mathematical Association of America, Washington, D.C., 1980. MR 588510
E. J. McShane, A unified theory of integration, Amer. Math. Monthly 80 (1973), 349–359. MR 318434, DOI 10.2307/2319078

Theory of functions of a real variable

The Riemann-Stieltjes approach to integration

W. F. Pfeffer, The generalized Riemann integral in higher dimensions, Measure theory and its applications (Sherbrooke, Que., 1982) Lecture Notes in Math., vol. 1033, Springer, Berlin, 1983, pp. 269–275. MR 729542, DOI 10.1007/BFb0099864
W. F. Pfeffer, The divergence theorem, Trans. Amer. Math. Soc. 295 (1986), no. 2, 665–685. MR 833702, DOI 10.1090/S0002-9947-1986-0833702-0

12

Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578