Institute for Mathematical Science
Stony Brook University
office: Math Tower 4-103
phone: (631) 632-8266
e-mail: remus.radu@stonybrook.edu
The PDF version of the schedule is available for print here.
| Date | Topic | Reading | Assignments |
| Aug 30 | Introduction: basic definitions and notations Null sets |
1, 2.1 | |
| Sept 1 | The Cantor middle-thirds set Outer measure |
2.1, 2.2 | HW1 (Due Sept 8) Solutions |
| Sept 6 | no class (Labor day) | ||
| Sept 8 | σ-algebras Properties of measurable sets |
2.3, 2.4 | HW2 (Due Sept 20) Solutions |
| Sept 13 | Lebesgue measure & properties | 2.4, 2.5, 2.7 | |
| Sept 15 | Borel sets Regularity of Lebesgue measure |
2.5, 2.7 | |
| Sept 20 | Example of a non measurable set Lebesgue-measurable functions |
Appendix, 3.1 | HW3 (Due Sept 29) Solutions |
| Sept 22 | Measurable functions | 3.1, 3.2 | |
| Sept 27 | Properties of measurable functions & examples | 3.3, 3.4, 3.6 | |
| Sept 29 | The Lebesgue integral | 4.1, 4.2, 4.8 | HW4 (Due Oct 6) Solutions |
| Oct 4 | Integrable functions Modes of convergence; Egorov's Theorem |
4.2, 4.3, 8.1 | |
| Oct 6 | Fatou's Lemma The Monotone Convergence Theorem |
4.2, 4.8 | HW5 (Due Oct 20) Solutions |
| Oct 11 | The Dominated Convergence Theorem | 4.3, 4.4 | |
| Oct 13 | Relations to the Riemann integral Applications of the convergence theorems |
4.4, 4.5 | |
| Oct 18 | Approximation of measurable functions | 4.6, 4.8 | |
| Oct 20 | Spaces of integrable functions: introduction | 5.1 | HW6 (Due Nov 3) Solutions |
| Oct 25 | Midterm (10:00-11:20am) Covers: 1, 2.1-2.5, 2.7, 3.1-3.4, 3.6, 4.1-4.6, 4.8
-- Solutions
Practice Midterm FA2014 & Solutions. |
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| Oct 27 | The space L1; completeness | 5.2 | |
| Nov 1 | Inner product spaces The Hilbert space L2 |
5.2, 5.5 | |
| Nov 3 | Lp spaces | 5.3, 5.5 | HW7 (Due Nov 10) Solutions |
| Nov 8 | Lp spaces & examples The L∞ space |
5.3 | |
| Nov 10 | Multi-dimensional Lebesgue measure | 6.1, 6.2 Video |
HW8 (Due Nov 22) Solutions |
| Nov 15 | Construction of the product measure | 6.3 | |
| Nov 17 | Fubini's Theorem & applications | 6.4, 6.6 | |
| Nov 22 | Abstract measure theory: absolutely continuous measures, singular measures, examples |
7.1, 7.2 | HW9 (Due Nov 29) Solutions |
| Nov 24 | no class (Thanksgiving) | ||
| Nov 29 | The Radon-Nikodym Theorem I | 7.2, 7.5 | HW10 (Due Dec 8) Solutions |
| Dec 1 | The Radon-Nikodym Theorem II | 7.2, 7.3 | |
| Dec 6 | Lebesgue-Stieltjes measures & FTC | 7.2 | |
| Dec 8 | Lebesgue decomposition theorem Review |
7.3 | |
| Dec 16 | Final Exam (11:15am-1:45pm) -- cumulative, covers everything
Practice Final FA2014 & Solutions. |
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