Difference between revisions of "Math 641: Functions of a Real Variable"
From MathWiki
(→Minimal learning outcomes) |
(→Minimal learning outcomes) |
||
Line 46: | Line 46: | ||
#* Minkowski's Inequality | #* Minkowski's Inequality | ||
#* Completeness of L<sup>p</sup> | #* Completeness of L<sup>p</sup> | ||
− | # | + | # Measures on product spaces |
#* Tonelli Theorem | #* Tonelli Theorem | ||
#* Fubini Theorem | #* Fubini Theorem | ||
Line 54: | Line 54: | ||
#* Absolutely continuous functions | #* Absolutely continuous functions | ||
#* Integrating derivatives of absolutely continuous functions | #* Integrating derivatives of absolutely continuous functions | ||
− | # | + | # Measures on topological spaces |
#* Borel sets | #* Borel sets | ||
#* Convergence in measure | #* Convergence in measure |
Revision as of 10:08, 14 August 2008
Contents
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
- Abstract measure theory
- σ-algebras
- Measures
- Positive measures
- Signed measures
- σ-finite measures
- Complete measures
- Measurable spaces
- Measure spaces
- 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
- Lp spaces
- Hölder's Inequality
- Minkowski's Inequality
- Completeness of Lp
- Measures on product spaces
- Tonelli Theorem
- Fubini Theorem
- 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
- 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