Difference between revisions of "Math 641: Functions of a Real Variable"

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#** Absolutely continuous functions
 
#** Absolutely continuous functions
 
#** Integrating derivatives of absolutely continuous functions
 
#** Integrating derivatives of absolutely continuous functions
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Revision as of 10:37, 14 August 2008

Catalog Information

Title

Functions of Real and Complex Variables 1.

Credit Hours

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

Prerequisites

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. Operations on measures
    • Absolutely continuous measures
    • Mutually singular measures
    • Lebesgue Decomposition Theorem
    • Radon-Nikodym 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
    • Radon 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





Additional topics

Courses for which this course is prerequisite