TMV100 / MMA110 Integration theory Autumn 21

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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.

Information for first day of course

Presentation slides for Chapter 3 of lecture notes presentationschapter3.pdf

Presentation slides for Chapter 4 of lecture notes presentationschapter4.pdf

Presentation slides for Chapter 5 of lecture notes presentationschapter5.pdf

Presentation slides for Chapter 6 of lecture notes presentationschapter6.pdf

New version of chapter 6 slides

Presentation slides for Chapter 7 of lecture notes  presentationschapter7.pdf

new version of presentation chapter 7

Presentation slides for Chapter 8 of lecture notes presentationschapter8.pdf

updated presentation chapter 8

Presentation slides for Chapter 9 of lecture notes presentationschapter9.pdf

 

Presentation slides for Section 3.15 (additive and linear functions) presentationsadditive.pdf

 

Program

The schedule of the course is in TimeEdit.

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).

September 6 homework

September 9 homework

exercises 16 sept

ruin 1 page proof of Exercise 1.33

sept 20 exercises

sept 23 exercises

Notes on infinite product measures

oct 7 exercises

Oct 8 notes on covering lemma

oct 14 notes..motivation background on total variation

exercises oct 18 (last day)

 

Day Sections Content
8/30

JS 3.1-3.3

F pp 19-22

JJ 3.1,3.2,3.5

Algebras, sigma-algebras, d-systems, Dynkin's Lemma, measure spaces
9/2

JS 3.3-3.7

F 1.3,4

JJ 3.7 and Theorem 3.10

Measure spaces, outer measure, Caratheodory's Theorem,

Construction  of Lebesgue measure,  uniqueness, nononmeasurable sets
9/3

JS  3.9-3.12

 Cantor function, Distribution functions,  Borel-Cantelli
9/6

 

 Exercises, Review, discussion
9/9

JS 4.1-4.3

F 2.1,2 up to (but not including) Theorem 2.14.

JJ 5 until Theorem 5.6

 Measurable functions, integration of non-negative functions.
9/10

JS 4.2-4.3

F Theorems 2.14-2.20

F 2.3 until p 55

Monotone convergence theorem,

integration of  real valued functions,  Fatou's lemma, Lebesgue dominated convergence
9/13 Exercises, Review, discussion
9/16

JS 4.4-4.5

F 2.4

Modes of convergence,  Some inequalities (Markov and

Chebyshev)
9/17

JS 5.1-5.5

F p 22-23 and F2.5

 Product sigma algebras, product measures, Fubini-Tonelli Theorem,  certain counterexamples
9/20 Continuation with product measures
9/23 Exercises, Review, discussion
9/24

JS 6.1-6.3

JJ p. 16, pp 21-22, 24-25

 Random variables, expectation (putting probability theory inside measure and integration theory)
9/27

JS 6.1-6.3

JJ p 9-10, 26-27 (until 4.3), Theorem 8.2, 8.3

 Borel-Cantelli lemmas, weak and strong law of large numbers
9/30 Exercises, Review, discussion
10/1

JS 7.1

F 3.1

Signed measures, Jordan-Hahn decomposition theorems,

mutual singularity
10/4

JS 7.2-7.3

F 3.2

Absolute continuity,

Radon-Nikodym theorem,Lebesgue’s decomposition theory
10/7 Exercises, Review, discussion
10/8

JS 8.1-8.4

F pp.95-96

 3-times covering lemma, Hardy-Littlewood maximal function, maximal theorem
10/11

JS 8.5

F pp.97-100

 

 Lebesgue’s differentiation theorem
10/14 Exercises, Review, discussion
10/15

JS 9.1-9.2

F pp. 101-107

Functions of bounded variation, absolute

continuity, Fundamental theorem of calculus

10/18

JS 9.1-9.2

F pp. 101-107

 Continuation with the above
10/21 Exercises, Review, discussion
10/22 Possible Review 

 

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Recommended exercises:  (1) Those contained in the lecture notes, (2) exercises

from Folland below and (3)  exercises from the supplementary exercise sheet (if you have time).

You should prioritise (1) and (2).

Chapter Exercises
Folland 1

2, 3, 4, 8, 9, 10, 18(a,b), 19,23, 28, 30, 31, 33
Folland 2

1, 3, 4, 5, 6, 8, 9, 13, 14, 16, 19, 21, 22, 25, 28(a,b), 30, 33, 37, 38, 47, 48
Folland 3

4, 5, 6, 9, 11, 17 (but only if you are interested in probability), 22, 25, 30, 31, 32, 39, 40, 41, 42(d)

supplementary exercise sheet (these might or might not be  updated as the course proceeds)

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Course summary:

Date Details Due