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
2 - A Refresher of Topology and Ordinal Numbers
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 is a brief reminder of point-set topology including examples of the most prominent topologies needed later on in the text. Further topics include ordinal numbers and the ordinal space (as a topological space), cardinality and counting and the construction of the Cantor middle-thirds set and the Cantor function (devil’s staircase) and its inverse function.
Keywords
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- Information
- Counterexamples in Measure and Integration , pp. 36 - 54Publisher: Cambridge University PressPrint publication year: 2021