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

From MathWiki
Jump to: navigation, search
(Minimal learning outcomes)
(Minimal learning outcomes)
Line 60: Line 60:
 
#* Radon measures
 
#* Radon measures
 
#* 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
 
#* Convergence in measure
 
#* Convergence in measure
 
#* Hahn Decomposition Theorem
 
#* Hahn Decomposition Theorem
Line 66: Line 67:
 
#* Mutually singular measures
 
#* Mutually singular measures
 
#* Lebesgue Decomposition Theorem
 
#* Lebesgue Decomposition Theorem
#* Lusin's Theorem
 
 
#* Egorov's Theorem
 
#* Egorov's Theorem
 
# Lebesgue measure on <b>R</b><sup>n</sup>
 
# Lebesgue measure on <b>R</b><sup>n</sup>
#* Mapping properties of Lebesgue measure
+
#* Existence
 +
#* Composition with affine maps
 
#* Differentiation on <b>R</b> and integration
 
#* Differentiation on <b>R</b> and integration
 
#** Derivative of integral is the integrand a.e.
 
#** Derivative of integral is the integrand a.e.

Revision as of 09:45, 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
    • Density of Cc in Lp
  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
    • Convergence in measure
    • Hahn Decomposition Theorem
    • Jordan Decomposition Theorem
    • Radon-Nikodym Theorem
    • Mutually singular measures
    • Lebesgue Decomposition Theorem
    • Egorov'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