Lecturer (and assistant)

• Silvius Klein
  
• Office: room 902, Sentralbygg 2
  
• Email: silviusk [at] math [dot] ntnu [dot] no
  
• Phone: 735 93695
  
• Office hours: Thursday 17-18 in S23 or by appointment.

Lectures

• Tuesday 10:15-12 in room MA23.
• Thursday 12:15-14 in room F3.

Exercise session

• Thursday 16:15-17 in room S23.

Textbook

• McDonald & Weiss: A Course in Real Analysis, second edition. This is the official textbook for the class, as it contains all the topics in the syllabus.

• However, we will also use other sources, as follows:

1. Some of the lecture notes written by the previous lecturer, Eugenia Malinnikova. They are available here, scroll down to the bottom of the page.
2. The lecture notes on probabilities available on Terence Tao's blog.
3. The following notes on ergodic theory and dynamical systems, written by Omri Sarig, plus the paper of I. Katznelson and B. Weiss, "A simple proof of some ergodic theorems", published in the Israel Journal of Mathematics 42 (1982), no. 4, 291–296.

• The following are standard textbooks, which one could always use as reference for corresponding topics in analysis:

1. For measure theory & topics in probability theory (plus some topology and functional analysis)–Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications.
2. For probability theory–Rick Durrett, Probability: Theory and Examples (freely available here).
3. For ergodic theory–Peter Walters, An Introduction to Ergodic Theory. There are many other books or notes online.

• You should make an effort to take good notes in class. In the end, they will be your most relevant reference for the homework assignments and for the final exam.

Syllabus

In this course we will present some topics in measure theory, probabilities, dynamical systems and ergodic theory. Here is a tentative list of concepts to be covered.

• On measure theory:

1. Signed measures; Hahn's decomposition theorem; Jordan's decomposition theorem.
2. The Radon-Nikodym theorem; Lebesgue's decomposition theorem of measures; distribution functions of Borel measures on the real line.
3. The Riesz–Markov–Kakutani representation theorem.
4. (Haussdorf measures—not covered).

• On probability theory:

1. Foundations of probability theory: the mathematical model of probability.
2. Random variables, integration and expectation.
3. Product measures and independence.
4. The weak and strong law of large numbers.
5. The central limit theorem.

• On dynamical systems and ergodic theory.

1. The definition of a dynamical system. Examples of dynamical systems.
2. Poincaré's recurrence theorem.
3. Ergodicity, mixing and other kinds of randomness.
4. The pointwise ergodic theorem of Birkhoff.
5. Some applications to number theory.

Reference group members

• Emilie Arentz-Hansen emilieba [at] stud [dot] ntnu [dot] no
• August Peter B. Sonne apsonne [at] stud [dot] ntnu [dot] no

Examination

There will be homework assigned regularly throughout the semester. Some of it will be graded. The grade received on your homework will count 20% towards your final grade.

There will be an oral exam at the end of the semester, which will count the remaining 80% towards your final grade for the course.