629 - Introduction to Measure and Integration
Math 522, or consent of instructor.
Students planning further studies in analysis, probability, or statistics.
This is an introduction to measure and integration theory. It is particularly suitable for further studies in analysis, probability and statistics.
- Lebesgue measure on the line: outer measure, measurable sets, nonmeasurable sets, measurable functions.
- Lebesgue integration on the line.
- Monotone convergence theorem, Fatou's Lemma, dominate convergence theorem.
- Almost everywhere convergence, convergence in measure, Egoroff's theorem.
- Differentiation, absolute continuity, derivatives of integrals.
- General measure and integration theory.
- Signed measures, Hahn decomposition theorem, Jordan decomposition.
- Radon-Nikodym theorem, Lebesgue decomposition.
- Outer measure, extension of measures, Lebesgue-Stieltjes measures.
- Product measures, Fubini and Tonelli theorems.
- Probability: conditional probability and expectation, distribution functions, statistical independence. (Optional)