| Instructor: Office: E-mail: |
Neil
Lyall Boyd GSRC 602A "my last name"@math.uga.edu |
Time/Location: Problem Session (8105): Office Hours: |
TR
11:00-12:15 Boyd 326 W 2:30-3:30 Boyd 326 TR 12:15-1:00 & W 5:00-6:00 |
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| Principal
Textbook: Measure and
Integral,
by R. L. Wheeden and A. Zygmund (Secondary Textbook: Real Analysis, by E. M. Stein and R. Shakarchi) |
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| Syllabus:
We will
cover chapters 3-10, and some additional
topics |
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| Handouts:
may have appeared here,
but didn't |
| Homework/Exams: |
Homework will be assigned regularly | |
| Midterm Exam 1: Tuesday the 7th of October (in class) | ||
| Midterm Exam 2: Tuesday the 18th of November (in class) | ||
| Final Exam: Thursday the 11th of December (12:00-3:00) | ||
| Grading:
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Homework:
30% (challenge questions
5%) |
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| Midterms:
20%
(each) |
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| Final: 30% |
| Homework 1 (Due Thursday the 4th of September) |
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| Homework 2 (Due Thursday the 11th of September) | |
| Homework 3 (Due Thursday the 18th of September) | |
| Homework 4 (Due Thursday the 25th of September) | |
| Homework 5 (Due Thursday the 2nd of October) | |
| Homework 6 (Due Thursday the 16th of October) | |
| Homework 7 (Due Thursday the 23rd of October) | |
| Homework 8 (Due Thursday the 30th of October) | |
| Homework 9 (Due Thursday the 13th of November) | |
| Homework 10 (Due Thursday the 4th of December) |
| Preliminaries 1 Week |
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| The
Riemann integral and its limitations Can we assign a "measure" to all subsets of the real line? |
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| (Whitney's)
decomposition theorm for open sets The Cantor set & Cantor-Lebesgue function |
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| Lebesgue
Measurable Sets 2 Weeks |
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| Lebesgue
outer measure and its properties Definition of Lebesgue measurable sets and their Lebesgue measure |
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| The
measurable sets form a sigma-algebra Corollary: F-sigma and G-delta set characterization of measurability |
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| Lebesgue
measure (on the measurable sets) is countably-additive Corollary: Behaviour of monotone sequences of sets |
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| Invariance
properties of Lebesgue measure Linear transformations acting on measurable sets (quantitative effect) |
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| The difference set of a set of
positive
measure contains an interval Corollary: Non-measurable sets (revisited) |
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| Lebesgue Measurable Functions 2 Weeks |
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| Properties of measurable functions Littlewood's three principles: (i) A measurable set is nearly an open set (ii) A measurable function is nearly a continuous function (Lusin's theorem) (iii) A convergent seq of measurable functions is nearly uniformly convergent (Egorov's theorem) |
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| Convergence in measure Egorov's Theorem |
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| Approximation
by simple functions Lusin's Theorem |
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| The Lebesgue Integral 3 Weeks |
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| The
integral of non-negative functions: Basic
properties |
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| The integral of non-negative functions: Convergence theorems | ||
| The
integral of arbitrary measurable functions |
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| The
space of integrable functions is a Banach space Different modes of convergence |
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| EXAM 1 |
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| Continuous
functions with compact support are dense in L^1 Invariance properties of the Lebesgue integral |
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| Comparison with the Riemann integral (Lebesgue's) Criterion for Riemann integrability |
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|
Repeated
Integration
1.5 Weeks |
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| Fubini's theorem & Tonelli's theorem |
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| Convolutions
and the Fourier Transform |
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|
Differentiation
1.5 Weeks
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| Lebesgue differentiation theorem Hardy-Littlewood maximal function theorem |
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| Extensions
and generalizations |
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| Indefinite integrals and absolute
continuity (at the moment statements only?) |
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|
L^p
Spaces
1 Week |
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| Banach
space properties of L^p spaces |
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| Further properties of L^p spaces | ||
| Hilbert Spaces 2 Weeks |
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| Inner
products Closed subspaces & orthogonal projections |
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| Linear functionals, dual spaces Riesz representation theorem (Hilbert spaces only) |
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| EXAM 2 | ||
| Orthogonality and Fourier series |
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| Abstract Integration 1 Week |
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| Abstract measure and integration |
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| Radon-Nikodym
theorem |
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