Students planning further studies in analysis, probability, or statistics.
Math 629 is an exploration of Lebesgue measure and integration and general measure theory. This course is suitable for advanced undergraduates in mathematics and graduate students in departments outside mathematics, and is fundamental to much of graduate analysis and statistics. Math 521 (first-semester real analysis) or its equivalent will be assumed.
This is an introduction to measure and integration theory. It is particularly suitable for further studies in analysis, probability and statistics.
Graduate courses in the subject
- 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)