Course syllabus

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Course PM

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.

 

The student representatives for the course are

Gustav Gille (gustavgi"at"chalmers.se)  and 
Rasmus Grönlund (rasmus.gronlund"at"gmail.com).

Feel free to contact them.

Information for the exam

List of proofs of theorems you should know

Choodle information and some more oral exam information (same as in announcement)

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).

exercises for Monday's cancelled class

 

Day Sections Content
9/1

JS 1, 2 and 3.1-3.2

Overview, background and motivation. Algebras, sigma-algebras, d-systems, Dynkin's Lemma, measure spaces
9/4

JS 3.3-3.7

Measure spaces, outer measure, Caratheodory's Theorem,

Construction of Lebesgue measure,  uniqueness, nononmeasurable sets
9/5

JS  3.3-3.7

Measure spaces, outer measure, Caratheodory's Theorem,

Construction  of Lebesgue measure,  uniqueness, nononmeasurable sets
9/8

JS 3.9-3.12

 Distribution functions,  the Cantor set, the Cantor function and the Cantor measure, Borel-Cantelli Lemma
9/11

 

 Exercises, Review, discussion
9/12 JS 4.1-4.3 Measurable functions, integration of non-negative functions
9/15 JS 4.2-4.3

Monotone convergence theorem, integration of  real valued functions,  Fatou's lemma, Lebesgue dominated convergence
9/18

 

Exercises, Review, discussion
9/19

JS 4.4-4.5

Modes of convergence,  Some inequalities (Markov and

Chebyshev)
9/22 JS 5.1-5.5 Product sigma algebras, product measures, Fubini-Tonelli Theorem,  certain counterexamples
9/25 Continuation with product measures
9/26

 

 Exercises, Review, discussion
9/29

JS 6.1-6.3

 Random variables, expectation (putting probability theory inside measure and integration theory)
10/2 JS 6.1-6.3 Borel-Cantelli lemmas, weak and strong law of large numbers
10/3

 

Exercises, Review, discussion
10/6

JS 7.1

Signed measures, Jordan-Hahn decomposition theorems,

mutual singularity
10/9 JS 7.2-7.3 Absolute continuity, Radon-Nikodym theorem,Lebesgue’s decomposition theory
10/10

 

 Exercises, Review, discussion
10/13 JS 8.1-8.4 3-times covering lemma, Hardy-Littlewood maximal function, maximal theorem
10/16 JS 8.5-8.8 Lebesgue’s differentiation theorem
10/17

 

Exercises, Review, discussion
10/20

JS 9.1-9.2

Functions of bounded variation, absolute

continuity, Fundamental theorem of calculus
10/23 JS 9.1-9.3 Continuation with the above
10/24 Exercises, Review, discussion

 

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Recommended exercises: 

Chapter 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