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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#* Dominated Convergence Theorem
 
#* Dominated Convergence Theorem
 
#* Effect of sets of measure zero
 
#* Effect of sets of measure zero
 +
#* Absolutely continuous measures
 +
#* Mutually singular measures
 +
#* Lebesgue Decomposition Theorem
 +
#* Radon-Nikodym Theorem
 +
#* Hahn Decomposition Theorem
 +
#* Jordan Decomposition Theorem
 
# L<sup>p</sup> spaces
 
# L<sup>p</sup> spaces
 
#* H&#246;lder's Inequality
 
#* H&#246;lder's Inequality
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#* Riesz Representation Theorem (for positive linear functionals on C<sub>c</sub>)
 
#* Riesz Representation Theorem (for positive linear functionals on C<sub>c</sub>)
 
#* Lusin's Theorem
 
#* Lusin's Theorem
#* Convergence in measure
 
#* Hahn Decomposition Theorem
 
#* Jordan Decomposition Theorem
 
#* Radon-Nikodym Theorem
 
#* Mutually singular measures
 
#* Lebesgue Decomposition Theorem
 
#* Egorov's Theorem
 
 
# Lebesgue measure on <b>R</b><sup>n</sup>
 
# Lebesgue measure on <b>R</b><sup>n</sup>
 
#* Existence
 
#* Existence

Revision as of 10:09, 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
    • Absolutely continuous measures
    • Mutually singular measures
    • Lebesgue Decomposition Theorem
    • Radon-Nikodym Theorem
    • Hahn Decomposition Theorem
    • Jordan Decomposition Theorem
  3. Lp spaces
    • Hölder's Inequality
    • Minkowski's Inequality
    • Completeness of Lp
    • Density of Cc in Lp
    • Convergence in norm
    • Almost-everywhere convergence
    • Almost-uniform convergence
    • Convergence in measure
    • Egoroff's Theorem
  4. Measures on product spaces
    • Tonelli Theorem
    • Fubini Theorem
  5. 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
  6. Lebesgue measure on Rn
    • Existence
    • Composition with affine maps
    • 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

Additional topics

Courses for which this course is prerequisite