Table of contents

Measure Theory from First Principles

A rigorous path from sets and Lebesgue measure to integration, convergence, product measures, and Radon–Nikodym theory

Read each section in order. Every title can be opened as a TheoryTrace document.

• Cover
• Copyright
• How to read this book
• Introduction
• Chapter 1: Sets, Logic, and Countability
• Chapter 2: Why Length Needs a Theory
• Chapter 3: Algebras, Sigma-Algebras, and Measurable Spaces
• Chapter 4: Measures and Their Basic Properties
• Chapter 5: Outer Measure and Carathéodory’s Criterion
• Chapter 6: Lebesgue Measure on the Real Line
• Chapter 7: Structure of Measurable Sets
• Chapter 8: Nonmeasurable Sets and the Limits of Size
• Chapter 9: Measurable Functions
• Chapter 10: Convergence of Measurable Functions
• Chapter 11: The Lebesgue Integral for Nonnegative Functions
• Chapter 12: Integrable Functions and the Dominated Convergence Theorem
• Chapter 13: Comparing Riemann and Lebesgue Integration
• Chapter 14: Modes of Convergence and Integration
• Chapter 15: Lp Spaces
• Chapter 16: Product Measures
• Chapter 17: Fubini’s and Tonelli’s Theorems
• Chapter 18: Signed and Complex Measures
• Chapter 19: The Radon–Nikodym Theorem
• Chapter 20: Differentiation of Measures on the Real Line
• Chapter 21: Measure Theory in Probability
• Chapter 22: Regular Borel Measures and Weak Convergence
• Chapter 23: Applications and Problem-Solving Patterns
• Chapter 24: A Bridge to Modern Analysis
• Conclusion