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
1 - A Panorama of Lebesgue Integration
Published online by Cambridge University Press: 27 May 2021
- 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 a brief non-technical overview on Lebesgue’s integration theory with respect to an abstract measure. Further topics include key theorems from the theory, such as convergence theorems (dominated convergence, monotone convergence, Fatou’s lemma), product measures (Fubini’s and Tonelli’s theorem) and abstract differentiation (Radon--Nikodým theorem).
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- Counterexamples in Measure and Integration , pp. 1 - 35Publisher: Cambridge University PressPrint publication year: 2021