# MA3105 Advanced real analysis, Spring 2014

**10.04** The date of the oral examination is **May 28** from 10.15 to 12.15 send me an email to get your time slot. We meet on Monday **May 19, 10.15 in room 734** for questions. The final list of topics for the exam is below

**20.01** Lecture on Tuesday is at 10.15, usual room

**15.01** This week we discuss the laws of large numbers, lecture tomorrow is at 12.15 in 734.

**08.01** We started yesterday with Borel-Cantelli lemmas and Infinite Monkey Theorem (ch.5.1 in McDonald, Weiss). Next lecture is on Thursday **12.15 room 734 SB II**.

**20.12** First lecture will be given on January 7th, Tuesday, room R81 (10.15)

We discuss schedule and the course and start our topic for the week which is *foundation of probability theory from analytical point of view*. We will learn the zero-one law and prepare the ground for the law of large numbers scheduled for the next week.

### Lectures

Tuesday 10.15-12.00, R81

Thursday 12.15-14.00, to be announced

Lecturer: Eugenia Malinnikova

room 948, Sentralbygg II

phone 73550257

e-mail eugenia [at] math [dot] ntnu [dot] no

### Preliminary Curriculum

Signed measures and the Radon-Nykodym theorem.

Riesz-Markov theorem, the dual of C(X).

Mathematical model of probability, The Law of Large Numbers.

Dynamical systems and Ergodic Theorem.

Hausdorff measures and dimension

Fourier transform and applications.

We will mostly use the textbook:

McDonald, Weiss,* A course in Real Analysis*, Academic press.

More detailed plan of the lectures and final list of topics: List of topics

### Lecture notes (updated)

week 2 LN 1 : Foundations of the probability theory, zero-one law

week 3-4 LN 2: Laws of large numbers, Shannon's theorem

week 5 LN 3: Signed measures

week 6 LN 4: Radon-Nykodim theorem; representation theorem

week 7 LN 5: Hausdorff measures and self-similar sets

week 8 Riesz Representation theorem

week 9 LN 6: Central Limit Theorem

weeks 9-10 Probabilistic number theory, slides

week 11 LN 7: Dynamical systems, Ergodicity

week 12-13 LN 8: Entropy of a dynamical system, Kolmogorov-Sinai theorem