Math 641: Functions of a Real Variable

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Catalog Information


Functions of Real and Complex Variables 1.

Credit Hours



Math 542 or instructor's consent


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


Minimal learning outcomes

  1. Abstract measure theory
    • σ-algebras
    • Measures
      • Positive measures
      • Signed measures
      • σ-finite 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
    • Monotone Convergence Theorem
    • Fatou's Lemma
    • Dominated Convergence Theorem
    • Effect of sets of measure zero
  3. Lp spaces
    • Hölder's Inequality
    • Minkowski's Inequality
    • Completeness of Lp
  4. Product Measures
  5. Differentiation and integration
  6. Miscellaneous
    • Borel sets

Additional topics

Courses for which this course is prerequisite