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Keywords:
Henstock-Kurzweil integral; bounded linear functional; bounded variation
Summary:
Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.
References:
[1] Alexiewicz, A.: Linear functionals on Denjoy-integrable functions. Colloq. Math. 1 (1948), 289-293. DOI 10.4064/cm-1-4-289-293 | MR 0030120 | Zbl 0037.32302
[2] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics Vol. 4, AMS (1994). DOI 10.1090/gsm/004/09 | MR 1288751 | Zbl 0807.26004
[3] Hildebrandt, T. H., Schoenberg, I. J.: On linear functional operations and the moment problem for a finite interval in one or several dimensions. The Annals of Mathematics (2) 34 317-328. MR 1503109 | Zbl 0006.40204
[4] Kreyszig, Erwin: Introductory Functional Analysis with Applications. John Wiley & Sons, New York-London-Sydney (1978). MR 0467220 | Zbl 0368.46014
[5] Kurzweil, J.: On multiplication of Perron integrable functions. Czech. Math. J 23 (1973), 542-566. MR 0335705 | Zbl 0269.26007
[6] Peng-Yee, Lee: Lanzhou Lectures on Henstock Integration. World Scientific (1989). MR 1050957
[7] Peng-Yee, Lee, Výborný, R.: The integral, An Easy Approach after Kurzweil and Henstock. Australian Mathematical Society Lecture Series 14 (Cambridge University Press, 2000). MR 1756319
[8] Tuo-Yeong, Lee, Tuan-Seng, Chew, Peng-Yee, Lee: Characterisation of multipliers for the double Henstock integrals. Bull. Austral. Math Soc. 54 (1996), 441-449. DOI 10.1017/S0004972700021857 | MR 1419607
[9] Tuo-Yeong, Lee, Tuan-Seng, Chew, Peng-Yee, Lee: On Henstock integrability in Euclidean spaces. Real Anal. Exchange 22 (1996/97), 382-389. MR 1433623
[10] Tuo-Yeong, Lee: Multipliers for some non-absolute integrals in the Euclidean spaces. Real Anal. Exchange 24 (1998/99), 149-160. MR 1691742
[11] Tuo-Yeong, Lee: A full descriptive definition of the Henstock-Kurzweil integral in the Euclidean space. Proc. London Math. Soc. 87 (2003), 677-700. MR 2005879
[12] Tuo-Yeong, Lee: A full characterization of multipliers for the strong $\rho$-integral in the Euclidean space. Czech. Math. J. 54 (2004), 657-674. DOI 10.1007/s10587-004-6415-7 | MR 2086723
[13] Tuo-Yeong, Lee: Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. J. Math. Anal. Appl. 298 (2004), 677-692. DOI 10.1016/j.jmaa.2004.05.033 | MR 2086983
[14] Tuo-Yeong, Lee: A characterisation of multipliers for the Henstock-Kurzweil integral. Math. Proc. Cambridge Philos. Soc. 138 (2005), 487-492. DOI 10.1017/S030500410500839X | MR 2138575
[15] Tuo-Yeong, Lee: Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral II. J. Math. Anal. Appl. 323 (2006), 741-745. DOI 10.1016/j.jmaa.2005.10.045 | MR 2262241
[16] Tuo-Yeong, Lee: A multidimensional integration by parts formula for the Henstock-Kurzweil integral. Math. Bohem. 133 (2008), 63-74. MR 2400151
[17] Liu, G. Q.: The dual of the Henstock-Kurzweil space. Real Anal. Exchange 22 (1996/97), 105-121. MR 1433600 | Zbl 0879.26046
[18] Mikusiński, Piotr, Ostaszewski, K.: The space of Henstock integrable functions II. In New integrals, (P. S. Bullen, P. Y. Lee, J. L. Mawhin, P. Muldowney and W. F. Pfeffer, Editors), Lecture Notes in Math. 1419 (Springer-Verlag, Berlin, Heideberg, New York 1990) 136-149. MR 1051926
[19] Ostaszewski, K. M.: The space of Henstock integrable functions of two variables. Internat. J. Math. and Math. Sci. 11 (1988), 15-22. DOI 10.1155/S0161171288000043 | MR 0918213 | Zbl 0662.26003
[20] Sargent, W. L. C.: On the integrability of a product. J. London Math. Soc. 23 (1948), 28-34. DOI 10.1112/jlms/s1-23.1.28 | MR 0026113 | Zbl 0031.29201
[21] Sargent, W. L. C.: On linear functionals in spaces of conditionally integrable functions. Quart. J. Math., Oxford Ser. 1 (1950), 288-298. DOI 10.1093/qmath/1.1.288 | MR 0039919 | Zbl 0039.11801
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