Difference between revisions of "Math 641: Functions of a Real Variable"
From MathWiki
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#* Tonelli Theorem | #* Tonelli Theorem | ||
#* Fubini Theorem | #* Fubini Theorem | ||
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# Measures on topological spaces | # Measures on topological spaces | ||
#* Borel sets | #* Borel sets | ||
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#* Jordan Decomposition Theorem | #* Jordan Decomposition Theorem | ||
#* Radon-Nikodym Theorem | #* Radon-Nikodym Theorem | ||
| − | #* Riesz Representation Theorem (for bounded linear | + | #* Riesz Representation Theorem (for bounded linear functionals on L<sup>p</sup>) |
#* Mutually singular measures | #* Mutually singular measures | ||
#* Lebesgue Decomposition Theorem | #* Lebesgue Decomposition Theorem | ||
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#* Lusin's 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 | ||
| + | #* 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 | ||
</div> | </div> | ||
Revision as of 10:14, 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
- Measures on topological spaces
- Borel sets
- Convergence in measure
- Hahn Decomposition Theorem
- Jordan Decomposition Theorem
- Radon-Nikodym Theorem
- Riesz Representation Theorem (for bounded linear functionals on Lp)
- Mutually singular measures
- Lebesgue Decomposition Theorem
- Lusin's Theorem
- Egorov's Theorem
- Lebesgue measure on Rn
- Mapping properties of Lebesgue measure
- 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
