Vak
2024-2025

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

In MyTimetable, you can find all course and programme schedules, allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.

Questions? Watch the video, read the instructions, or contact the ISSC helpdesk.

Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.

## 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%).

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

As a student, you are responsible for enrolling on time through MyStudyMap.

In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.
Extensive information about the operation of MyStudyMap can be found here.

There are two enrolment periods per year:

• Enrolment for the fall opens in July

• Enrolment for the spring opens in December

Note:

• It is mandatory to enrol for all activities of a course that you are going to follow.

• Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.

• Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.

## Contact

Lecturer: Dr. S.C. Hille (shille[at]math.leidenuniv.nl)
Teaching assistants: see Brightspace pages of the course.

## Remarks

Software
Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.