Book contents
- Frontmatter
- Contents
- Preface
- User’s Guide
- List of Topics and Phenomena
- 1 A Panorama of Lebesgue Integration
- 2 A Refresher of Topology and Ordinal Numbers
- 3 Riemann Is Not Enough
- 4 Families of Sets
- 5 Set Functions and Measures
- 6 Range and Support of a Measure
- 7 Measurable and Non-Measurable Sets
- 8 Measurable Maps and Functions
- 9 Inner and Outer Measure
- 10 Integrable Functions
- 11 Modes of Convergence
- 12 Convergence Theorems
- 13 Continuity and a.e. Continuity
- 14 Integration and Differentiation
- 15 Measurability on Product Spaces
- 16 Product Measures
- 17 Radon–Nikodým and Related Results
- 18 Function Spaces
- 19 Convergence of Measures
- References
- Index
3 - Riemann Is Not Enough
Published online by Cambridge University Press: 27 May 2021
Book contents
- Frontmatter
- Contents
- Preface
- User’s Guide
- List of Topics and Phenomena
- 1 A Panorama of Lebesgue Integration
- 2 A Refresher of Topology and Ordinal Numbers
- 3 Riemann Is Not Enough
- 4 Families of Sets
- 5 Set Functions and Measures
- 6 Range and Support of a Measure
- 7 Measurable and Non-Measurable Sets
- 8 Measurable Maps and Functions
- 9 Inner and Outer Measure
- 10 Integrable Functions
- 11 Modes of Convergence
- 12 Convergence Theorems
- 13 Continuity and a.e. Continuity
- 14 Integration and Differentiation
- 15 Measurability on Product Spaces
- 16 Product Measures
- 17 Radon–Nikodým and Related Results
- 18 Function Spaces
- 19 Convergence of Measures
- References
- Index
Summary
This chapter contains counterexamples showing the shortcomings of Riemann integration.
- Type
- Chapter
- Information
- Counterexamples in Measure and Integration , pp. 55 - 72Publisher: Cambridge University PressPrint publication year: 2021