Course syllabus

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This page contains the program of the course: lectures  and suggested exercises. Other information, such as learning outcomes, teachers, literature, lecture notes and examination information, are in a separate course PM.

Program

The schedule of the course is in TimeEdit.

last lecture

 

Information for the exam

 

Lectures (the correspondence between what will be covered and the days is very approximate: in particular, the days with planned exercises might not be the days we do the exercises).

 

 

Day Sections Content
9/2

JS 3.1-3.3

Algebras, sigma-algebras, d-systems, Dynkin's Lemma, measure spaces
9/5

JS 3.3-3.7

Measure spaces, outer measure, Caratheodory's Theorem,

Construction  of Lebesgue measure,  uniqueness, nononmeasurable sets
9/6

JS  3.9-3.12

Distribution functions,  the Cantor set, the Cantor function and the Cantor measure, Borel-Cantelli Lemma
9/9

 

 Exercises, Review, discussion
9/12

JS 4.1-4.3

 Measurable functions, integration of non-negative functions.
9/13 NO CLASS NO CLASS
9/16 JS 4.2-4.3

Monotone convergence theorem, integration of  real valued functions,  Fatou's lemma, Lebesgue dominated convergence
9/19

 

Exercises, Review, discussion
9/20

JS 4.4-4.5

Modes of convergence,  Some inequalities (Markov and

Chebyshev)
9/23 JS 5.1-5.5 Product sigma algebras, product measures, Fubini-Tonelli Theorem,  certain counterexamples
9/26 Continuation with product measures
9/27

 

 Exercises, Review, discussion
9/30

JS 6.1-6.3

 Random variables, expectation (putting probability theory inside measure and integration theory)
10/3 JS 6.1-6.3 Borel-Cantelli lemmas, weak and strong law of large numbers
10/4

 

Exercises, Review, discussion
10/7

JS 7.1

Signed measures, Jordan-Hahn decomposition theorems,

mutual singularity
10/10 JS 7.2-7.3 Absolute continuity, Radon-Nikodym theorem,Lebesgue’s decomposition theory
10/11

 

 Exercises, Review, discussion
10/14 JS 8.1-8.4 3-times covering lemma, Hardy-Littlewood maximal function, maximal theorem
10/17 JS 8.5 Lebesgue’s differentiation theorem
10/18

 

Exercises, Review, discussion
10/21

JS 9.1-9.2

Functions of bounded variation, absolute

continuity, Fundamental theorem of calculus
10/24 JS 9.1-9.2 Continuation with the above
10/25 Exercises, Review, discussion

 

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Recommended exercises: 

Chapter Exercises

One should do as many of the exercises in the notes as you have time for.  Here, nonetheless, are some recommended exercises, a number of which I will present.
3

4,9(b),10 (3),11,12,16,18, 29, 35,36
4

1, 7, 9, 10, 12, 21, 22, 23, ,25, 27, 29
5

3,4
6

2,4
7

3,4,6,7,12,13, (16 for those interested in probability theory)
8

2, 3
9

1,2 (and 5 if you want a challenge)

 

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Course summary:

Course Summary
Date Details Due