TMV101 / MMA110 Integration theory Autumn 25

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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

 

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
9/1

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 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 9.1-9.2 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:

Date Details Due