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
      • 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. Lp spaces
    • Hölder's Inequality
    • Minkowski's Inequality
    • Completeness of Lp
  4. Measures on product spaces
    • Tonelli Theorem
    • Fubini Theorem
  5. Differentiation on R and integration
    • Derivative of integral is the integrand a.e.
    • Functions of bounded variation
    • Absolutely continuous functions
    • Integrating derivatives of absolutely continuous functions
  6. Measures on topological spaces
    • Borel sets
    • Convergence in measure
    • Hahn Decomposition Theorem
    • Jordan Decomposition Theorem
    • Radon-Nikodym Theorem
    • Riesz Representation Theorem (for bounded linear functions on Lp)
    • Mutually singular measures
    • Lebesgue Decomposition Theorem
    • Lebesgue measure
      • Mapping properties of Lebesgue measure
    • Lusin's Theorem
    • Egorov's Theorem
  7. Measures on Rn

Additional topics

Courses for which this course is prerequisite