×

zbMATH — the first resource for mathematics

A measure-theoretic characterization of the Henstock-Kurzweil integral revisited. (English) Zbl 1174.26005
Summary: We show that the measure generated by the indefinite Henstock-Kurzweil integral is \(F_{\sigma \delta }\) regular. As a result, we give a shorter proof of the measure-theoretic characterization of the Henstock-Kurzweil integral.

MSC:
26A39 Denjoy and Perron integrals, other special integrals
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] D. Bongiorno, L. Di Piazza and V. A. Skvortsov: Variational measures related to local systems and the ward property of \( \mathcal{P} \) -adic path bases. Czech. Math. J. 56 (2006), 559–578. · Zbl 1164.26316 · doi:10.1007/s10587-006-0037-1
[2] L. Di Piazza: Variational measures in the theory of the integration in \(\mathbb{R}\)m. Czech. Math. J. 51 (2001), 95–110. · Zbl 1079.28500 · doi:10.1023/A:1013705821657
[3] Claude-Alain Faure: A descriptive definition of some multidimensional gauge integrals. Czech. Math. J. 45 (1995), 549–562. · Zbl 0852.26010
[4] R. Henstock, P. Muldowney and V. A. Skvortsov: Partitioning infinite-dimensional spaces for generalized Riemann integration. Bull. London Math. Soc. 38 (2006), 795–803. · Zbl 1117.28010 · doi:10.1112/S0024609306018819
[5] J. Kurzweil and J. JarnĂ­k: Differentiability and integrability in n dimensions with respect to \(\alpha\)-regular intervals. Results Math. 21 (1992), 138–151. · Zbl 0764.28005
[6] Lee Tuo-Yeong: A full descriptive definition of the Henstock-Kurzweil integral in the Euclidean space. Proc. London Math. Soc. 87 (2003), 677–700. · Zbl 1047.26006 · doi:10.1112/S0024611503014163
[7] Lee Tuo-Yeong: Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. J. Math. Anal. Appl. 298 (2004), 677–692. · Zbl 1065.26013 · doi:10.1016/j.jmaa.2004.05.033
[8] Lee Tuo-Yeong: A characterisation of multipliers for the Henstock-Kurzweil integral. Math. Proc. Cambridge Philos. Soc. 138 (2005), 487–492. · Zbl 1078.28004 · doi:10.1017/S030500410500839X
[9] Lee Tuo-Yeong: The Henstock variational measure, Baire functions and a problem of Henstock. Rocky Mountain J. Math 35 (2005), 1981–1997. · Zbl 1099.26009 · doi:10.1216/rmjm/1181069626
[10] Lee Tuo-Yeong: Some full descriptive characterizations of the Henstock-Kurzweil integral in the Euclidean space. Czech. Math. J. 55 (2005), 625–637. · Zbl 1081.26008 · doi:10.1007/s10587-005-0050-9
[11] Lee Tuo-Yeong: Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral II. J. Math. Anal. Appl. 323 (2006), 741–745. · Zbl 1107.26011 · doi:10.1016/j.jmaa.2005.10.045
[12] B. S. Thomson: Derivates of interval functions. Mem. Amer. Math. Soc. 93 (1991). · Zbl 0734.26003
[13] A. J. Ward: On the derivation of additive interval functions of intervals in m-dimensional space. Fund. Math. 28 (1937), 265–279. · JFM 63.0192.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.