# Math 641: Functions of a Real Variable

### Title

Functions of Real and Complex Variables 1.

3

### Prerequisite

Math 542 or instructor's consent

### Description

Fundamentals of measure and integration, Borel measures, product measures, L^ spaces, introduction to functional analysis, Radon Nikodym theorem, differentiation theory, Fourier transforms.

## Desired Learning Outcomes

Math 641 is a course in abstract measure and integration theory. The official current course description mentions a couple of items that are not envisioned to be core topics in the renovated course: functional analysis and Fourier transforms. A new 500-level course in linear functional analysis would subsume functional-analytic material previously taught in Math 641, and Fourier transforms will be covered in Math 541 and not repeated in Math 641. 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

It is proposed that the current prerequisite course, Math 542, be deleted from the catalog and therefore as a prerequisite for Math 641. We probably should strongly recommend that students take Math 541 as a prerequisite. At a minimum, students should have had a course equivalent to Math 316.

### 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-algebra
• 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