Math 629 - Introduction to Measure and Integration

• Prerequisites: Math 522.
• Frequency: Spring (II)
• Student Body: students planning further studies in analysis, probability, or statistics.
• Credits: 3. (N-A)
• Recent Texts: Royden, Real Analysis. Ash, Real Analysis and Probability.
• Course Coordinator: Daniel Rider
• 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

Content coverage:

• 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-spaces
• Probability: conditional probability and expectation, distribution functions, statistical independence. (Optional)