## 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)

## Mode of instruction

Lectures (2 hours per week)

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

## Assessment method

(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.

## 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).
Bogachev

Draft chapters of a book on the subject will be provided during the course

```
Lecture notes
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## Registration

You have to sign up for classes and examinations in uSis.

## Contact information

Lecturer: Dr. S.C. Hille (shille@math.leidenuniv.nl)

Teaching assistants: see Brightspace pages of the course.