## Admission requirements

Students must have completed 'Inleiding Maattheorie’ (4082INMT3) or a comparable course on the fundamentals of measure and integration theory. Acquaintance with the basics of point set topology and metric spaces, and normed vector spaces (e.g. through the course ‘'Linear Analysis') is needed. The book by Cohn 'Measure Theory' provides ample background material.

## Description

The course starts by introducing and studying additional structures on the set of finite measures. For example, they constitute a convex cone that can be embedded in a vector space: the signed measures. This is an ordered vector space with natural norm(s) defined on it. This order structure relates to the so-called Hahn-Jordan decomposition of signed measures. Absolute continuity of measures and the Radon-Nikodym Theorem are discussed.

The core of the course considers Borel measures on topological spaces, mainly locally compact Hausdorff or separable complete metric spaces (Polish spaces). Various regularity concepts for (signed) measures are introduced. The Riesz Representation Theorem is proven, that identifies the dual space of continuous functions on a locally compact Hausdorff space, vanishing at infinity., with the particular class of signed Radon measures.

Considering Borel measures on non-locally compact base spaces leads to various mathematical complications. In the course we focus on the case when the underlying space is Polish (i.e. metrisable, becoming a separable complete metric space), which is a common assumption in Analysis and Probability Theory. We discuss weak convergence of measures and the associated Dudley metric, which is defined by a norm on the signed measures. This introduces a weaker norm (and topology) than that related to the order structure. It is a highly useful concept, e.g. in Probability Theory. Important are relative compactness results for sets of measures: uniform tightness of measures and the Prokhorov Theorem.

The topological structure enables discussion of dynamics in spaces of measures. We provide examples of those defined by so-called Markov operators and one-parameter semigroups of such operators. Important concepts are: invariant (probability) measures and ergodic measures, the existence (Krylov-Bogolyubov Theorem), possible uniqueness and stability of invariant measures and conditions for that.

## Course objectives

The course introduces students to more advanced topics in measure and integration theory, such as norms and weak (vector space) topologies on the vector space of signed measures. Understanding of these concepts allows her/him to consider applications to Dynamical Systems and Markov processes. This provides a good starting point for further study, either in the direction of Analysis (e.g. equations in spaces of measures) or Probability Theory (e.g. Markov processes)

## Timetable

You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.

MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).

For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different timetables into one. This video explains how to do this.

## Mode of instruction

Lectures (2 hours per week)

Three take-home assignments with exercises, organized per topic discussed

## Assessment method

The final grade of the course is computed by weighted average from two components:

1. three take-home individual assignments (practicals, equally weighted average; 25%)

2. written exam (75%)

A retake exam is oral, over a selection of topics from the course material. The final grade in case of a retake exam is simply the mark for the retake exam (100%).

## Reading list

The course combines well-established results with those that are recent developments in the field of Analysis and Probability Theory. Thus, not a single book can and will be used. Detailed Lecture Notes will be provided with ample references to the literature. Recommended books (but not mandatory):

On fundamentals of measure theory: Donald L. Cohn, Measure Theory ISBN: 978-1-4614-6955-1 (Print) 978-1-4614-6956-8 (Online) (available as e-book via Leiden University Library).

Encyclopaedic, on topics of the course and beyond: V.I. Bogachev, Measure Theory, Volume 1 and 2, Berlin: Springer-Verlag, 2007

See further the references in the Lecture Notes (made available through Brightspace).

## Registration

From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.

Please note that it is compulsory to both preregister and confirm your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Confirming your exam participation is possible until ten days before the exam.

Extensive FAQ's on MyStudymap can be found here.

## Contact

Lecturer: Dr. S.C. Hille (shille[at]math.leidenuniv.nl)

Teaching assistants: see Brightspace pages of the course.