Difference between revisions of "Math 641: Functions of a Real Variable"
From MathWiki
(→Minimal learning outcomes) |
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| Line 25: | Line 25: | ||
#** Signed measures | #** Signed measures | ||
#** σ-finite measures | #** σ-finite measures | ||
| + | #** Complete measures | ||
#* Measurable spaces | #* Measurable spaces | ||
#* Measure spaces | #* Measure spaces | ||
| Line 33: | Line 34: | ||
#* 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 | ||
| Line 42: | Line 47: | ||
#* 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 | ||
</div> | </div> | ||
Revision as of 09:43, 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
- Product Measures
- 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
- 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
