Difference between revisions of "Math 541: Real Analysis"
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<div style="-moz-column-count:2; column-count:2;"> | <div style="-moz-column-count:2; column-count:2;"> | ||
− | + | # Lebesgue measure on <b>R</b><sup>n</sup> | |
+ | #* Inner and outer measures | ||
+ | #* Construction of Lebesgue measure | ||
+ | #* Properties of Lebesgue measure | ||
+ | #** Effect of basic set operations | ||
+ | #** Limiting properties | ||
+ | #** Its domain | ||
+ | #** Approximation properties | ||
+ | #** Sets of outer measure zero | ||
+ | #** Invariance w.r.t. isometries | ||
+ | #** Effect of dilations | ||
+ | #* Existence of nonmeasurable sets | ||
+ | # Lebesgue integration on <b>R</b><sup>n</sup> | ||
+ | #* Measurable functions | ||
+ | #* Simple functions | ||
+ | #* Approximation of measurable functions with simple functions | ||
+ | #* The extended reals | ||
+ | #* Integrating nonnegative functions | ||
+ | #* Integrating absolutely-integrable functions | ||
+ | #* Integrating on measurable sets | ||
+ | #* Basic properties of the Lebesgue integral | ||
+ | #** Linearity | ||
+ | #** Monotonicity | ||
+ | #** Effects of sets of measure zero | ||
+ | #** Absolute continuty of integration | ||
+ | #** Fatou's Lemma | ||
+ | #** Monotone Convergence Theorem | ||
+ | #** Dominated Convergence Theorem | ||
+ | #** Differentiation w.r.t. a parameter | ||
+ | #** Linear changes of variable | ||
+ | #** Compatibility with Riemann integration | ||
+ | # Fubini's Theorem for <b>R</b><sup>n</sup> | ||
+ | # L<sup>1</sup>, L<sup>2</sup>, and L<sup>∞</sup> | ||
+ | #* Their completeness | ||
+ | #* Approximation by smooth functions | ||
</div> | </div> | ||
Revision as of 10:42, 13 August 2008
Contents
Catalog Information
Title
Real Analysis.
Credit Hours
3
Prerequisite
Description
Rigorous treatment of differentiation and integration theory; Lebesque measure; Banach spaces.
Desired Learning Outcomes
Prerequisites
Minimal learning outcomes
- Lebesgue measure on Rn
- Inner and outer measures
- Construction of Lebesgue measure
- Properties of Lebesgue measure
- Effect of basic set operations
- Limiting properties
- Its domain
- Approximation properties
- Sets of outer measure zero
- Invariance w.r.t. isometries
- Effect of dilations
- Existence of nonmeasurable sets
- Lebesgue integration on Rn
- Measurable functions
- Simple functions
- Approximation of measurable functions with simple functions
- The extended reals
- Integrating nonnegative functions
- Integrating absolutely-integrable functions
- Integrating on measurable sets
- Basic properties of the Lebesgue integral
- Linearity
- Monotonicity
- Effects of sets of measure zero
- Absolute continuty of integration
- Fatou's Lemma
- Monotone Convergence Theorem
- Dominated Convergence Theorem
- Differentiation w.r.t. a parameter
- Linear changes of variable
- Compatibility with Riemann integration
- Fubini's Theorem for Rn
- L1, L2, and L∞
- Their completeness
- Approximation by smooth functions