Henstock-Kurzweil integration on Euclidean spaces /

The Henstock-Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral more than 50 years ago. This relatively new integral is known to be equivalent to the classical Perron integral; in particular, it includes the...

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Bibliographic Details
Main Author: Lee, Tuo Yeong, 1967-
Format: eBook
Published: New Jersey : World Scientific, ©2011.
Series:Series in real analysis ; v. 12.
Online Access:CONNECT
Table of Contents:
  • 1. The one-dimensional Henstock-Kurzweil integral. 1.1. Introduction and Cousin's lemma. 1.2. Definition of the Henstock-Kurzweil integral. 1.3. Simple properties. 1.4. Saks-Henstock lemma. 1.5. Notes and remarks
  • 2. The multiple Henstock-Kurzweil integral. 2.1. Preliminaries. 2.2. The Henstock-Kurzweil integral. 2.3. Simple properties. 2.4. Saks-Henstock lemma. 2.5. Fubini's theorem. 2.6. Notes and remarks
  • 3. Lebesgue integrable functions. 3.1. Introduction. 3.2. Some convergence theorems for Lebesgue integrals. 3.3. [symbol]-measurable sets. 3.4. A characterization of [symbol]-measurable sets. 3.5. [symbol]-measurable functions. 3.6. Vitali covering theorem. 3.7. Further properties of Lebesgue integrable functions. 3.8. The L[symbol] spaces. 3.9. Lebesgue's criterion for Riemann integrability. 3.10. Some characterizations of Lebesgue integrable functions. 3.11. Some results concerning one-dimensional Lebesgue integral. 3.12. Notes and remarks
  • 4. Further properties of Henstock-Kurzweil integrable functions. 4.1. A necessary condition for Henstock-Kurzweil integrability. 4.2. A result of Kurzweil and Jarnik. 4.3. Some necessary and sufficient conditions for Henstock-Kurzweil integrability. 4.4. Harnack extension for one-dimensional Henstock-Kurzweil integrals. 4.5. Other results concerning one-dimensional Henstock-Kurzweil integral. 4.6. Notes and remarks
  • 5. The Henstock variational measure. 5.1. Lebesgue outer measure. 5.2. Basic properties of the Henstock variational measure. 5.3. Another characterization of Lebesgue integrable functions. 5.4. A result of Kurzweil and Jarnik revisited. 5.5. A measure-theoretic characterization of the Henstock-Kurzweil integral. 5.6. Product variational measures. 5.7. Notes and remarks.
  • 6. Multipliers for the Henstock-Kurzweil integral. 6.1. One-dimensional integration by parts. 6.2. On functions of bounded variation in the sense of Vitali. 6.3. The m-dimensional Riemann-Stieltjes integral. 6.4. A multiple integration by parts for the Henstock-Kurzweil integral. 6.5. Kurzweil's multiple integration by parts formula for the Henstock-Kurzweil integral. 6.6. Riesz representation theorems. 6.7. Characterization of multipliers for the Henstock-Kurzweil integral. 6.8. A Banach-Steinhaus theorem for the space of Henstock-Kurzweil integrable functions. 6.9. Notes and remarks
  • 7. Some selected topics in trigonometric series. 7.1. A generalized Dirichlet test. 7.2. Fourier series. 7.3. Some examples of Fourier series. 7.4. Some Lebesgue integrability theorems for trigonometric series. 7.5. Boas' results. 7.6. On a result of Hardy and Littlewood concerning Fourier series. 7.7. Notes and remarks
  • 8. Some applications of the Henstock-Kurzweil integral to double trigonometric series. 8.1. Regularly convergent double series. 8.2. Double Fourier series. 8.3. Some examples of double Fourier series. 8.4. A Lebesgue integrability theorem for double cosine series. 8.5. A Lebesgue integrability theorem for double sine series. 8.6. A convergence theorem for Henstock-Kurzweil integrals. 8.7. Applications to double Fourier series. 8.8. Another convergence theorem for Henstock-Kurzweil integrals. 8.9. A two-dimensional analogue of Boas' theorem. 8.10. A convergence theorem for double sine series. 8.11. Some open problems. 8.12. Notes and remarks.