| Date | Topic | Reading | Assignments |
|---|---|---|---|
| Aug 27 | Syllabus Introduction: basic definitions and notations Null sets |
1, 2.1 | Homework 1 (Due Sept 5) Solutions |
| Aug 29 | The Cantor middle-thirds set
Outer measure |
2.1, 2.2 | |
| Sept 3 | Properties of outer measure
Properties of measurable sets |
2.2, 2.3 | Homework 2 (Due Sept 12) Solutions |
| Sept 5 | Properties of Lebesgue measure
\(\sigma\)-algebras |
2.4, 2.5 | |
| Sept 10 | Borel sets A non-measurable set |
2.5, 2.7, Appendix | Homework 3 (Due Sept 19) Solutions |
| Sept 12 | Measurable functions | 3.1, 3.2 | |
| Sept 17 | Properties of measurable functions Examples |
3.3, 3.4, 3.6 | Homework 4 (Due Sept 26) Solutions |
| Sept 19 | Properties of measurable functions Egorov's theorem |
3.3, 3.4, 3.6, Egorov's theorem | |
| Sept 24 | Simple functions The Lebesgue integral |
4.1 | Homework 5 (Due Oct 3) Solutions |
| Sept 26 | Integral of non-negative functions Monotone Convergence Theorem |
4.1, 4.2 | |
| Oct 1 | Properties of the Lebesgue integral Fatou's lemma |
4.2 | Homework 6 (Due Oct 10) Solutions |
| Oct 3 | Integrable functions The Dominated Convergence Theorem |
4.3, 4.4, 4.8 | |
| Oct 8 | Relations between Riemann and Lebesgue integrals Approximation of measurable functions |
4.5, 4.6, 4.8 | Homework 7 (Due Oct 17) Solutions |
| Oct 10 | Spaces of integrable functions The space \(L^1\) |
5.1 | |
| Oct 15 | Fall Break (no classes in session) | ||
| Oct 17 | Completeness of \(L^1\) The Hilbert space \(L^2\) |
5.1, 5.2 | |
| Oct 22 | Properies of \(L^2\) \(L^p\) spaces |
5.2, 5.3 | Homework 8 (Due Oct 31) Solutions |
| Oct 24 | Midterm (in class) | Cumulative (Everything covered up to the fall break) |
Practice exams from previous semesters:
Practice 1 (Solutions) Practice 2 (with solutions) Practice 3 (Solutions) Practice 4 (Solutions) Practice 5 Practice 6 |
| Oct 29 | \(L^p\) spaces The space \(L^\infty\) |
5.3 | Homework 9 (Due Nov 7) Solutions |
| Oct 31 | Multi-dimensional Lebesgue measure | 6.1, 6.2 | |
| Nov 5 | Fubini's theorem | 6.3, 6.4 | Homework 10 (Due Nov 14) Solutions |
| Nov 7 | Digression: linear change of coordinates in Lebesgue measure | ||
| Nov 12 | Applications of Fubini's theorem | 6.3, 6.4 | Homework 11 (Due Nov 21) Solutions |
| Nov 14 | Abstract measure theory | 7.2 | |
| Nov 19 | The Radon-Nikodym theorem | 7.2 | Homework 12 (Due Dec 3) Solutions |
| Nov 21 | The Radon-Nikodym theorem Lebesgue's decompostion theorem |
7.2 | |
| Nov 26 | Lebesgue-Stieltjes measures | 7.3 | |
| Nov 28 | Thanksgiving Break (no classes in session) | ||
| Dec 3 | Absolutely continuous functions Functions of bounded variation |
7.3 | |
| Dec 5 | Fundamental theorem of Calculus | 7.3 | |
| Dec 19 | Final exam Time: 8:00am-10:45am Location: E&S 181 (same as lectures) |
Cumulative |
Practice exams from previous semesters:
Practice 1 (Solutions) Practice 2 Practice 3 Practice 4 Pracitce 5 Practice 6 |
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