Henstock-Kurzweil Integration on Euclidean Spaces (Series in Real Analysis) (Series in Real Analysis, 12)

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 powerful Lebesgue integral. This book presents an introduction of the multiple Henstock-Kurzweil integral. Along with the classical results, this book contains some recent developments connected with measures, multiple integration by parts, and multiple Fourier series. The book can be understood with a prerequisite of advanced calculus. 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 One-dimensional Henstock-kurzweil Integral -- Multiple Henstock-kurzweil Integral -- Lebesgue Integrable Functions -- Further Properties Of Henstock-kurzweil Integrable Functions -- Henstock Variational Measure -- Multipliers For The Henstock-kurzweil Integral -- Some Selected Topics In Trigonometric Series -- Some Applications On The Henstock-kurzweil Integral To Double Trigonometric Series. Lee Tuo Yeong. Includes Bibliographical References (p. 295-303) And Index. The Henstock-Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral. This book presents an introduction of the multiple Henstock-Kurzweil integral. It covers the developments connected with measures, multiple integration by parts, and multiple Fourier series.