## ANNOUNCEMENTS

### Please Sign and Return Your FERPA Form!

Graded homework and quizzes will be returned to you in recitation section. If for some reason you do not collect your classwork in recitation section, you have the option to pick up your graded work in the Section boxes located on the 2nd floor of Sloan.

However, due to the Family Educational Rights and Privacy Act (FERPA) privacy regulations, we cannot place graded work in a public location, without your permission.

If didn't file a FERPA form last term, i.e., for Ma 108a, please fill out the form this term to indicate how you prefer your uncollected graded classwork to be handled. Remember that the first opportunity to collect your work is in your recitation section.

Download this form, and return it, separately and unstapled, with your first homework assignment.

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## COURSE DESCRIPTION

Math 108b will be principally concerned with developing the theory of Lebesgue measure. This deals with the fundamental question of how we should think of the size of a set of real numbers. Once we have this under control, we will develop a notion of integration which somewhat generalizes the Riemann integration which is taught in Math 1a. By studying Lebesgue theory in this way, in depth and from the ground up, we will gradually develop and understand some of the powerful techniques of analysis which can be used in almost any situation. In particular, we will study the Dominated convergence theorem and the Lebesgue differentiation theorem. We will discuss some of the far-reaching consequences of the techniques used in proving the Lebesgue differentiation theorem.

## OFFICE HOURS

Lukas Schimmer: Wednesday 3:00 - 4:00, 380 Sloan

Connor Meehan: Wednesday 5:00 - 6:00, 153 Sloan

Emad Nasrollahpoursamami: Tuesday 2:30-3:30, 160 Sloan

## POLICIES

### Grades

TBA

### Homework Policy

Homework is worth 30% of the grade. Each problem set is worth the same, except your homework with the lowest score, which will be dropped.

Problem sets are available before Saturday evening on the class web page. They should be turned in on Thursday at 4:00 PM in the Math department's drop box for Math 108b. They will be available for pick-up on Wednesday in the Math department's pick-up box for Math 108b. Sample solutions will be available on the webpage after the due date.

No late work will be accepted or extensions granted. (In exchange your lowest homework score will be dropped.)

You are encouraged to discuss homework problems with one another, however the solutions must be written up in your own words.

Due to federal privacy policies, each assignment is to be submitted with a cover sheet, which carries nothing other than the student's name (first & last name).

### Exam Policy

There are two exams, a midterm and a final each worth 35% of the grade. The respective dates and times will be announced. The final exam is cumulative. The midterm and final exams are open-book, timed, non-collaborative, the exact policy for which will be announced at that time.

## TOPICS COVERED

- Functions of bounded variation and the Riemann Stieltjes integral
- Lebesgue measure and Lebesgue outer measure
- Lebesgue measurable functions
- The Lebesgue integral
- Repeated integration
- Differentiation

## TEXTBOOKS

[1] Wheeden, R. L. and Zygmund, A.,* Measure and Integral: An Introduction
to Real Analysis, Second Edition*, ISBN 9781498702898

## LECTURE NOTES

Date | Description | Reference |
---|---|---|

01/04/17 | 1 dimonsional Riemann Integral, Higher dimensional Riemann
Integral, Motivation of Lebesgue Integral, Notation, Sets, Open and closed sets, Intervals |
Ma 1 [1] pp. 1-8 |

01/06/17 | Decomposing open sets, Discontinuities, Relative continuity, Functions of bounded variation, Examples | [1] pp. 8-18 |

01/09/17 | Positive/negative variation, Jordan's Theorem, Discontinuities | [1] pp. 19-22 |

01/11/17 | Variation as a limit, Variation as an integral, curves, graph of a curve, rectifiable curves | [1] pp. 22-25 |

01/13/17 | Riemann--Stieltjes integral, Integration by parts, Upper/Lower Riemann--Stieltjes sums | [1] pp. 26-31 |

01/16/17 | Martin Luther King Day | |

01/18/17 | Existence of Riemann--Stieltjes integrals, Mean-value theorem, Upper/Lower Riemann--Stieltjes sums, Lebesgue Outer Measure, Cantor set | [1] pp. 32-43 |

01/20/17 | Perfect sets are uncountable, Lebesgue--Cantor function, Invariance against rotation of coordinate axes, Lebesgue measurable sets, Lebesgue measure | [1] p. 10 & pp. 43-46 |

01/23/17 | Closed sets are measurable, Measurable sets form a Sigma Algebra, Borel sets, Countable additivity | [1] pp. 46-51 |

01/25/17 | Regularity of the Lebesgue measure, Characterizations of measurability, Lipschitz transformations | [1] pp. 51-55 |

01/27/17 | Lipschitz transformations, Linear transformations, Nonmeasurable sets | [1] pp. 55-61 |

01/30/17 | Lebesgue measurable functions, Properties of Lebesgue measurable functions | [1] pp. 61-66 |

02/01/17 | Properties of Lebesgue measurable functions Simple functions, Semicontinuous functions | [1] pp. 66-70 |

02/03/17 (Midterms) | Semicontinuous functions, Egorov's theorem, Lusin's theorem | [1] pp. 70-73 |

02/06/17 (Midterms) | Convergence in measure, Lebesgue integral for non-negative fucntions | [1] pp. 77-81 |

02/08/17 (Midterms) | Properties of the Lebesgue integral, existence, monotone convergence theorem | [1] pp. 82-84 |

02/10/17 | Linearity of the Lebesgue integral, Fatou's lemma, Lebesgue's dominated convergence theorem | [1] pp. 84-90 |

02/13/17 | Lebesgue integral for arbitrary measurable functions, Linearity, Convergence theorems | [1] pp. 90-96 |

02/15/17 | Lebesgue's dominated convergence theorem, Relation between Riemannâ€“Stieltjes and Lebesgue Integrals, Distribution function, Equimeasurable functions, | [1] pp. 96-101 |

02/17/17 | Lebesgue integral and partitions the range, Lp classes, Riemann and Lebesgue Integrals | [1] pp. 101-107 |

02/20/17 | President's Day | |

02/22/17 | Improper Riemann integrals and Lebesgue integrals, Riemann integrability, Fubini's theorem | [1] pp. 107-115 |

02/24/17 | Fubini's theorem, Tonelli's theorem | [1] pp. 115-119 |

02/27/17 | Tonelli's theorem, Applications of Fubini's/Tonelli's theorem | [1] pp. 119-124 |

03/01/17 | Marcinkiewicz theorem, Set functions, Indefinite integral, Lebesgue's Differentiation Theorem | [1] pp. 124-132 |

03/03/17 | Approximation of integrable functions, Simple Vitali Lemma, Hardy--Littlewood maximal function | [1] pp. 132-136 |

03/06/17 | Hardy--Littlewood lemma, Proof of Lebesgue's theorem | [1] pp. 136-139 |

03/08/17 | Extensions and corollaries of Lebesgue's theorem | [1] pp. 139-142 |

03/10/17 | Viali Covering lemma, Overview of absolutely continuyous functions and differentiation | [1] pp. 142-152 |

## HOMEWORK

Due Date | Homework | Solutions |
---|---|---|

01/12/17 | Problem set 1 | Solutions 1 |

01/19/17 | Problem set 2 | Solutions 2 |

01/26/17 | Problem set 3 | Solutions 3 |

02/02/17 | Problem set 4 | Solutions 4 |

02/16/17 | Problem set 5 | Solutions 5 |

02/23/17 | Problem set 6 | Solutions 6 |

03/02/17 | Problem set 7 | Solutions 7 |

03/09/17 | Problem set 8 |