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

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(Minimal learning outcomes)
(Minimal learning outcomes)
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#** Signed measures
 
#** Signed measures
 
#** σ-finite measures
 
#** σ-finite measures
 +
#** Complete measures
 
#* Measurable spaces
 
#* Measurable spaces
 
#* Measure spaces
 
#* Measure spaces
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#* Integrating nonnegative functions
 
#* Integrating nonnegative functions
 
#* Integrating L<sup>1</sup> functions
 
#* Integrating L<sup>1</sup> functions
 +
#* Integration on a measurable set
 +
#* Measures defined through integration
 +
#* Absolute continuity of integration
 +
#* Linearity of integration
 
#* Monotone Convergence Theorem
 
#* Monotone Convergence Theorem
 
#* Fatou's Lemma
 
#* Fatou's Lemma
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#* Completeness of L<sup>p</sup>
 
#* Completeness of L<sup>p</sup>
 
# Product Measures
 
# Product Measures
# Differentiation and integration
+
#* Tonelli Theorem
 +
#* Fubini Theorem
 +
# Differentiation on <b>R</b> and integration
 +
#* Derivative of integral is the integrand a.e.
 +
#* Functions of bounded variation
 +
#* Absolutely continuous functions
 +
#* Integrating derivatives of absolutely continuous functions
 
# Miscellaneous  
 
# Miscellaneous  
 
#* Borel sets
 
#* Borel sets
 +
#* Convergence in measure
 +
#* Hahn Decomposition Theorem
 +
#* Jordan Decomposition Theorem
 +
#* Radon-Nikodym Theorem
 +
#* Riesz Representation Theorem
 +
#* Mutually singular measures
 +
#* Lebesgue Decomposition Theorem
 +
#* Lebesgue measure
 +
#** Mapping properties of Lebesgue measure
 +
#* Lusin's Theorem
 +
#* Egorov's Theoprem
 
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</div>
  

Revision as of 08:43, 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. Lp spaces
    • Hölder's Inequality
    • Minkowski's Inequality
    • Completeness of Lp
  4. Product Measures
    • 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. Miscellaneous
    • Borel sets
    • Convergence in measure
    • Hahn Decomposition Theorem
    • Jordan Decomposition Theorem
    • Radon-Nikodym Theorem
    • Riesz Representation Theorem
    • Mutually singular measures
    • Lebesgue Decomposition Theorem
    • Lebesgue measure
      • Mapping properties of Lebesgue measure
    • Lusin's Theorem
    • Egorov's Theoprem

Additional topics

Courses for which this course is prerequisite