629 - Introduction to Measure and Integration
Prerequisites:
Math 522, or consent of instructor.
Frequency:
Spring (II)
Student Body:
Students planning further studies in analysis, probability, or statistics.
Credits:
3. (N-A) Recent Texts:
Folland Real analysis. Stein-Shakarchi, Real analysis.
Course Coordinator:
Betsy Stovall Background and Goals:
This is an introduction to measure and integration theory. It is particularly suitable for further studies in analysis, probability and statistics.
Alternatives:
n/a
Subsequent Courses:
Graduate courses in the subject Course Content:
- 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.
- L^p-spaces
- Probability: conditional probability and expectation, distribution functions, statistical independence. (Optional)