# MA3105 Advanced real analysis, Spring 2012

**13.03** One more schedule change, there will be a lecture on Tuesday, March 20th, but no lecture on Friday March 30th.

The last lecture is on April 20th, the oral exam will be arranged before June 8th.

**27.02** Extra lectures on Tuesday will be given on 6th and 13th of March, room 734 as usual, 14.15-16.00. No lectures next two Thursdays March 1st and 8th.

**23.02** There will be no lecture next Thursday, 01.03.

On Friday 02.03 we will discuss dynamical systems (problems from lecture note 6).

Drafts of the last lecture notes are below.

**12.01** First lecture is today at 14.15 in room 734.

Our topic for this week 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

Thursday 14.15-16.00, room 734, SB II

Friday 14.15-16.00, room 734, SB II

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: Preliminary plan

### Lecture notes

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

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

week 4 LN 3 : Signed measures, Radon-Nikodym theorem and conditional expectations

week 5 LN 4 : Decomposition of measures.

week 6 LN 5: Measurable Dynamical Systems, Pointwise Ergodic theorem

week 7 LN 6 : Entropy of dynamical system

week 8 LN 7 : Hausdorff measures

week 9 Examples of ergodic systems (presentations)

week 10 Riesz theorem in Hausdorff locally compact spaces (Rudin, Real and Complex analysis, ch. 2)

week 11 LN 8 : Central Limit Theorem

week 11 LN 9 : Central Limit Theorem and Law of the Iterated Logarithm

week 12 LN 10 : The Law of the Iterated Logarithm for dyadic martingales

week 13 Lattice random walk

week 14 Easter break

week 15 Probabilistic number theory: Hardy-Ramanujan and Erdös-Kac theorems