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The chapter and section numbers refer to the
course text: G. Folland, *Real Analysis: Modern Techniques
and Their Applications.*

Date | Topics |
Text | Assignments | |

1 | 9/3 | Important inequalities of Holder and Minkowski | Class Notes | Assignment due 9/10 Solutions, 1-8 |

2 | 9/8 | Metric spaces: examples, completeness and contraction mapping | Class Notes | |

3 | 9/10 | Compactness, Total Boundedness | Class Notes and Folland, 0.6 | Problems for 9/17 Solutions, 9-18 |

4 | 9/15 | The Arzela-Ascoli theorem | Class notes on Arzela-Ascoli | |

5 | 9/17 | Riemann-Stieltjes Integrals | Class Notes | Problems for 9/24 Solutions |

6 | 9/22 | Functions of bounded variation |
Class Notes | |

7 | 9/24 | Outline of the Lebesgue approach to integration Algebras and sigma-algebras |
Folland, Chapter 1 | Problems for 10/1 Solutions |

8 | 9/26 | Algebras and sigma algebras | Folland, Chapter 1 | |

9 | 10/1 | Measurable functions | Folland, Chapter 2, section 1 | Problems for 10/8 Solutions |

10 | 10/6 | Finitely additive measures, measures, elementary properites, extension problem |
Folland, Chapter 1 | |

11 | 10/8 | Caratheodory's extension theorem and applications | Folland, Chapter 1, section 4 |
Problems for 10/15 Solutions |

12 | 10/13 | Caratheodory's extension theorem and applications: Lebesgue measure Class notes on construction of measures |
Folland, Chapter 1 | |

13 | 10/15 | Caratheodory's extension theorem and applications: Lebesgue measure in d diimensions: a coin tossing space. |
Folland, Chapter 1 | Problems for 10/23 Solutions |

14 | 10/20 | Midterm | ||

15 | 10/23 | Properties of Lebesgue-Stieltjes measures; Non-measurable sets |
Folland, Section 1.5 | |

16 | 10/27 | Integration; definition and Fatou's lemma | Folland, Section 2.2 | |

17 | 10/29 | Monotone and dominated convergence theorems | Folland, Sections 2.2 and 2.3 | Problems for 11/5 Solutions |

18 | 11/3 | Relation between Riemann and Lebesgue integrals | Folland, Sections 2.2 and 2.3 | |

19 | 11/5 | Density of step functions, continuous functions; modes of convergence |
Folland, Sections 2.4 | Problems for 11/12 Solutions |

20 | 11/10 | Convergence in measure and a.e. convergence; Completelness of L^p; Change of variables |
Folland, Sections 2.4 | |

21 | 11/12 |
Change of variables; Product measure and the Fubini-Tonelli theorem |
Folland, Sections 2.5 | Problems for 11/19 Solutions |

22 | 11/17 | Product measure and the Fubini-Tonelli theorem, continued | Folland, Sections 2.5 | |

23 | 11/19 | Change of variable in Lebesgue integration; Signed measures |
Folland, Sections 2.6, 2.7, 3.1 | Problems for 12/3 Solutions |

24 | 12/1 | The Radon-Nikodym theorem | Folland, Section 3.2 | |

25 | 12/3 | Differentiation of Borel measures w.r.t. Lebesgue measure; Application to bounded variation functions |
Folland, Sections 3.3, 3.4 | Problems for 12/10 |

26-28 | 12/3 | Differentiation of Borel measures w.r.t. Lebesgue measure; Application to bounded variation functions |
Folland, Sections 3.3, 3.4, Class notes | |

29 | 12/16 | FINAL EXAM, 12-3, Hill 425 | Info on final (Please Read!) |