CHAPTER 21
Measure Theory and Advanced Integration
Why measure is needed
Length, area, probability and limitations of elementary integration.
σ-algebras and measures
Measurable sets and countable additivity.
Measurable functions
Preimages, simple functions and approximation.
Lebesgue integration
Integrating nonnegative and general measurable functions.
Convergence theorems
Monotone and dominated convergence and Fatou's lemma.
Connection to probability
Probability measures, expectation and foundations for advanced stochastic analysis.