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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...
| Clasificación: | Libro Electrónico |
|---|---|
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
WSPC
2011.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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.