#### Course information

`Lecturer (and assistant)`

- Office: room 902, Sentralbygg 2

- Phone: 735 93695

- Office hours: Thursday 17-18 in S23 or by appointment.

`Lectures`

`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:

- Some of the lecture notes written by the previous lecturer, Eugenia Malinnikova. They are available here, scroll down to the bottom of the page.
- The lecture notes on probabilities available on Terence Tao's blog.

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

- For measure theory & topics in probability theory (plus some topology and functional analysis)–Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications.
- For probability theory–Rick Durrett, Probability: Theory and Examples (freely available here).
- 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:

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

- On probability theory:

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

- On dynamical systems and ergodic theory.

- The definition of a dynamical system. Examples of dynamical systems.
- Poincaré's recurrence theorem.
- Ergodicity, mixing and other kinds of randomness.
- The pointwise ergodic theorem of Birkhoff.
- 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.