Ma 108b:  Classical Analysis (Winter 2016-17)


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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.


Math 108a


MWF 11:00 - 11:55, 151 Sloan.


Lukas Schimmer
Sloan 380


Connor Meehan

Emad Nasrollahpoursamami


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




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.



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


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 Riemann–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


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  


Midterm Solutions