# Math 641: Functions of a Real Variable

### Title

Functions of a Real Variable.

3

### Prerequisite

Math 541 or instructor's consent

### Description

Abstract measure and integration theory; L(p)(?) spaces; measures on topological and Euclidean spaces.

## Desired Learning Outcomes

Math 641 is a course in abstract measure and integration theory. There will be some repetition of topics between Math 541 and Math 641, but it is felt that the repetition will help solidify student understanding, and there will be a difference in approach, with the lower-level course taking a concrete approach restricted to Lebesgue measure.

### Prerequisites

We strongly recommend that students take Math 541 prior to Math 641.

### Minimal learning outcomes

Outlined below are topics that all successful Math 641 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

1. Abstract measure theory
• σ-algebras
• Measures
• Positive measures
• Signed measures
• σ-finite measures
• Complete measures
• Measurable spaces
• Measure spaces
2. Abstract integration theory
• Abstract measurable mappings
• Measurable real- and extended-real-valued functions
• Integrating simple functions
• Integrating nonnegative functions
• Integrating L1 functions
• Integration on a measurable set
• Measures defined through integration
• Absolute continuity of integration
• Linearity of integration
• Monotone Convergence Theorem
• Fatou's Lemma
• Dominated Convergence Theorem
• Effect of sets of measure zero
3. Operations on measures
• Absolutely continuous measures
• Mutually singular measures
• Lebesgue Decomposition Theorem
• Hahn Decomposition Theorem
• Jordan Decomposition Theorem
4. Lp spaces
• Hölder's Inequality
• Minkowski's Inequality
• Completeness of Lp
• Density of Cc in Lp
• Inclusion of Lp spaces
• Duality of Lp spaces
5. Convergence results
• Types of convergence
• Convergence in Lp-norm
• Almost-everywhere convergence
• Almost-uniform convergence
• Convergence in measure
• Relationships between different types of convergence
• Egoroff's Theorem
6. Measures on abstract product spaces
• Existence of product measure
• Tonelli's Theorem
• Fubini's Theorem
7. Measures on topological spaces
• Borel σ-algebras
• Locally compact Hausdorff spaces
• Urysohn's Lemma
• Partitions of unity
• Borel measures
• Locally finite measures
• Regular measures
• Riesz Representation Theorem (for positive linear functionals on Cc)
• Lusin's Theorem
8. Lebesgue measure on Rn
• Existence
• Composition with affine maps
• Change of variable formula for integration
• Differentiation and integration on R
• Derivative of integral is the integrand a.e.
• Functions of bounded variation
• Absolutely continuous functions
• Integrating derivatives of absolutely continuous functions