# MATH 6280 3.00AF (Fall 2013)Measure Theory

Announcements and documents will be posted here as they become available.

### Announcements

• The final exam (and comprehensive) was held Thursday Dec. 19, 10am-1pm in CB115. You were responsible for the topics covered in class, with the exception of conditional expectations and Hausdorff measures.
• Solutions to the 4th assignment are posted on the assignment page [linked to below].
• Textbook: Real Analysis by Royden and Fitzpatrick; 4th edition. The textbook is optional - you can follow the course purely using notes from class. I will pull some homework problems from the text (particularly optional ones), and will announce sections from the text so that people can read ahead if they wish.

### Topics

We covered the following topics:
• Definition of a measure, and basic properties (see sections 2.1 and 17.1 of Royden).
• Stated a theorem giving the existence of Lebesgue-Stieljes measures associated with increasing and right-continuous functions F. The proof has three steps:
• First, construct a pre-measure mu associated with F, on a concrete algebra of sets.
• Second, extend this pre-measure to an outer measure mu* defined on all sets.
• Third, show countable additivity when we restrict mu* to Borel sets (actually to what are called the mu*-measurable sets).
The theory this requires is covered in sections 2.2-2.3, 2.5, and 17.3-17.5 of Royden.
• Uniqueness of L-S measures, using monotone classes (not covered in Royden)
• The Cantor measure (section 2.7)
• Non-measurable sets (section 2.6)
• Approximation properties (section 2.4).
• Completions (section 17.1)
• Integration
• Measurable functions (sections 3.1, 3.2, 18.1)
• Integration of simple functions
• Integration of positive measurable functions (sections 4.2, 4.3, 18.2)
• Integrable functions (sections 4.4, 18.3)
• Monotone convergence, Fatou's lemma, Dominated convergence
• Convergence in measure and in Lp (sections 5.2, 7.1)
• Minkowski and Holder inequalities, completeness of Lp (sections 7.2, 7.3, 19.1)
• Product measure, Fubini and Tonelli (section 20.1)
• Signed measures, and the Jordan-Hahn decomposition (section 17.2)
• The Radon Nikodym theorem (section 18.4) and applications in probability (conditional expectations) and analysis.
• Hausdorff measures (you are not responsible for this topic)