# Introduction to Lebesgue Integration

## Section Information

 Section Time Location Instructor Office Hours (Cupples I, Room 17) 1 MWF 11:00 AM - 12:00 PM Duncker 101 Renato Feres Tu-Th 12:00 PM - 3:00 PM

 Please include [Math4121] in the subject line of any email message that pertains to this course. This will help avoid that I accidentally delete your message. My e-mail address is feres@math.wustl.edu.

## Text

Lebesgue Integration, 2nd Edition, by Soo Bong Chae
Springer 1995

## Topics to be covered

This course is an introduction to the theory of Lebesgue integration. Topics to be covered are:
• The Riemann integral;
• Lebesgue measure and integral on the real line;
• General measures, measurable functions;
• The main convergence theorems;
• Lebesgue spaces
• Measures on product spaces and Fubini's theorem
• Decomposition of measures
• Additional topics, as time permits, from: differentiation theory, measure theory based probability, Fourier analysis.

The plan is to have 10 weekly homework assignments, one midterm exam, and one final exam. The midterm exam will be in class, 50 minutes long, and its date is 3/03/2017 (during class time, in Duncker 101). It will be based on content covered in the homework sets. The final exam is on 5/09/2017 from 10:30AM to 12:30PM (place to be announced). The worst homework score will be dropped. Midterm and final exams will each have twice the weight of a homework assignment.

Therefore the final score will be determined as follows: (1) linearly scale each homework or exam score for a maximum of 100 points; (2) add the 9 best homework assignment scores plus twice the midterm exam score plus twice the final exam score and divide the total by 13.

The total score, out of 100, will be translated into a letter grade of A, B, C, D, F (with plus and minus shadings) in a way that is not harsher than indicated on the following table (if the grade distribution of the whole class is significantly lower than usual, then "curving" may be considered, but it is unlikely to be needed):

 Numerical Range Letter Grade [85, 100] A [70, 85) B [60, 70) C [50, 60) D [0, 50) F

Pluses and minuses are assigned so that each letter range is divided in three populations of roughly equal size. So, for example, the number of students getting A-, A, A+ will be approximately the same. Please note: this does not mean that A- corresponds to the interval [85, 90) or that A corresponds to [90, 95)! In particular, the exact grade cut-offs for the signs will not be known, to you as well as to me, until after the final exam.

## Homework

The plan is to have a total of 10 HW assignments. Of these, the 9 best scores will count towards the total.

Weekly assignments deadline is Friday 5:00PM. You should submit them using Crowdmark. (I'll have more to say about it on the first day of class.)

Under exceptional circumstances, I may accept them at a different time on Friday, but I plan on posting solutions on Saturday. Late assignment will not be accepted after solutions are posted.

You are both allowed and encouraged to collaborate on your assignments.

## Course plan

Please keep in mind that the plan given below is very tentative. We may fall behind (likely) or ahead (less likely) of the planned schedule on any given week.

 Week Textbook section covered Jan 18 - Jan 20 I.3, I.4 Jan 23 - Jan 27 Homework 1: 1/27 Solutions I.4, I.5, I.6 Jan 30 - Feb 03 Homework 2: 2/03 Solutions II.1, II.2, II.3 Feb 06 - Feb 10 Homework 3: 2/10 Solutions II.3, II.4, II.5 Feb 13 - Feb 17 Homework 4: 2/17 Solutions II.5, II.6 Feb 20 - Feb 24 Homework 5: 2/24 Solutions II.6, II.7 Feb 27 - Mar 03 Midterm: 3/03 Solutions III.1, III.2 Mar 06 - Mar 10 Homework 6: 3/10 Solutions III.3, III.4, III.5 Mar 20 - Mar 24 III.5, III.6, III.7 Mar 27 - Mar 31 Homework 7: 3/31 Solutions III.7, III.10 Apr 03 - Apr 07 Homework 8: 4/07 Solutions IV.1, IV.2 Apr 10 - Apr 14 Homework 9: 4/14 Solutions IV.3, IV.5 Apr 17 - Apr 21 IV.5 Apr 24 - Apr 28 Homework 10: 4/28 Solutions IV.5, IV.6 May 09 - Final Exam Final: 5/09 (10:30AM - 12:30PM, place to be announced) Solutions

Renato Feres
feres@math.wustl.edu
Cupples I, Room 17
(314) 935 - 6752 (office phone)

Department of Mathematics
Washington University
Campus Box 1146
Saint Louis
Missouri, 63130 USA.

Last Updated: January 11, 2017