TMV100 / MMA110 Integration theory Autumn 22

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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.

Information for the exam

Exercises for oral

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).

October 6 Lecture

 

Oct 7 lecture

 

October 10 lecture

 

 

Day Sections Content
8/29

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
9/1

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
9/2

JS  3.9-3.12

 Cantor function, Distribution functions,  Borel-Cantelli
9/5

 

 Exercises, Review, discussion
9/8

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.
9/9

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
9/12 Exercises, Review, discussion
9/15

JS 4.4-4.5

F 2.4

Modes of convergence,  Some inequalities (Markov and

Chebyshev)
9/16

JS 5.1-5.5

F p 22-23 and F2.5

 Product sigma algebras, product measures, Fubini-Tonelli Theorem,  certain counterexamples
9/19 Continuation with product measures
9/22 Exercises, Review, discussion
9/23

JS 6.1-6.3

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

 Random variables, expectation (putting probability theory inside measure and integration theory)
9/26

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
9/29 Exercises, Review, discussion
9/30

JS 7.1

F 3.1

Signed measures, Jordan-Hahn decomposition theorems,

mutual singularity
10/3

JS 7.2-7.3

F 3.2

Absolute continuity,

Radon-Nikodym theorem,Lebesgue’s decomposition theory
10/6 Exercises, Review, discussion
10/7

JS 8.1-8.4

F pp.95-96

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

JS 8.5

F pp.97-100

 

 Lebesgue’s differentiation theorem
10/13 Exercises, Review, discussion
10/14

JS 9.1-9.2

F pp. 101-107

Functions of bounded variation, absolute

continuity, Fundamental theorem of calculus

10/17

JS 9.1-9.2

F pp. 101-107

 Continuation with the above
10/20 Exercises, Review, discussion
10/21 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