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A Riemann-type theorem for a Riemann-type integral. (English) Zbl 1429.26011

Summary: For functions which are Henstock-Kurzweil integrable but not Lebesgue-integrable we prove a theorem which resembles the Riemann theorem on the rearrangement of conditionally convergent series.

MSC:

26A39 Denjoy and Perron integrals, other special integrals
40A10 Convergence and divergence of integrals
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References:

[1] Henstock, R.: A problem in two-dimensional integration. J. Austral. Math. Soc. Ser. A 35(3), 386-404 (1983) · Zbl 0549.26007 · doi:10.1017/S1446788700027087
[2] Lee, T.Y.: Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. J. Math. Anal. Appl. 298, 677-692 (2004) · Zbl 1065.26013 · doi:10.1016/j.jmaa.2004.05.033
[3] Pfeffer, W.: The Riemann Approach to Integration. Cambridge University Press, Cambridge (1993) · Zbl 0804.26005
[4] Riemann, B.: Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 13 (1867)
[5] Rao, G., Tulone, F.: Analogue of the Riemann-Dini theorem for non-absolutely convergent integrals. Le Matematiche, vol. LXII fasc. I, (2007) 129-134 · Zbl 1150.26004
[6] Rao, G., Tulone, F.: Henstock integral and Dini-Riemann theorem. Le Matematiche, vol. LXIV fasc. II, (2009) 71-77 · Zbl 1198.26015
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