Spaces of Measures

Based on a PhD course given by Sander Hille at the University of Milano in 2014

* Organisers*

Marcel de Jeu (mdejeu@math.leidenuniv.nl)

Sander Hille (shille@math.leidenuniv.nl)

Onno van Gaans (vangaans@math.leidenuniv.nl)

*Contact*

If you are interested please contact Marcel de Jeu.

*Topic*

The theory of measures acquires extra dimensions if the underlying point set is a topological space, and if various function spaces and corresponding weak topologies on spaces of measures are introduced. The classical Riesz representation theorem (identifying a space of measures on a locally compact Hausdorff space as a dual space) is one example of such an interplay between measure theory, topology and functional analysis, but there is much more that can be said. This seminar is concerned with these matters, coherently covering results that are scattered over the literature as well as recent research results.

The first part of the seminar deals with topologies that are used to topologise the vector space of finite Borel measures on a Polish space, its cone of positive measures and the subset of probability measures. An overview is provided of the relationships between these topologies, metrics that are used to metrise some of them, and functional analytic aspects of these spaces. Weak topologies, the Fortet-Mourier norm and various Wasserstein metrics will be introduced and compared.

The second part provides a brief introduction to Bochner integration in general, followed by highlighting the specific properties that hold for Bochner integration of measure-valued functions.

If time permits, in the third part the preceding topics are combined in the analysis of particular examples of dynamical systems in spaces of measures, e.g. as arising in measure-valued formulations of structured population models.

*Intended for*

Students, PhD students and faculty.

*Prerequisites*

Basic measure theory and topology, combined with some maturity in the functional analytic language. The national functional analysis course is more than enough preparation for the latter.

*Literature*

Sander Hille's lecture notes will be made available, and some other sources will be used as well. Here is a list of relevant literature.

*Venue*

Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building), Leiden. For the lecture rooms, see the detailed schedule below.

*Dates and time*

Friday afternoons, 14.00-17.00hr at the latest, on:

5 February

12 February

19 February

4 March

11 March

1 April

8 April

22 April

13 May

20 May

27 May

3 June

10 June

*EC*

6 EC for participation and delivering an afternoon filling lecture.

*Grade*

It is enevitable that some topics are more suitable for an attractive presentation than others, so, as in previous years, there will be no grades but simply a "pass".

*Please note*

If you are a student, but not from Leiden, contact your study advisor about the eligibility of this seminar for your own programme beforehand, in order to prevent unwanted surprises. In past years, there have always been non-Leiden participating students. If your institute should require this, then, although this is not the preferred method, a grade could be supplied instead of a "pass".

**Programme**

*Lecture 1:* 5 February 2016 (room B03): Sander Hile

Introduction and motivation; regularity

*Lecture 2:* 12 February 2016 (room 401) : Hent van Imhoff

Spaces of Lipschitz functions; molecular measures I

*Lecture 3:* 19 February 2016 (room 401): Maja Ziemlanska

Measures as functionals; molecular measures II; introduction to the Daniell integral

*Lecture 4:* 4 March 2016 (room 401): Sander Hille

Functionals defining measures I

*Lecture 5:* 11 March 2016 (room 401): Marcel de Jeu

Functionals defining measures II

*Lecture 6:* 1 April 2016 (room 408): Onno van Gaans

Sets of measures that are bounded in the total variation norm

*Lecture 7:* 8 April 2016 (room 408): Benthen Zeegers

Introduction to the Bochner integral; integration of measure-valued functions

*Lecture 8:* 22 April 2016 (room 408): Bart van Ginkel

A concrete example: measure-valued mass evolution

*Lecture 9:* 13 May 2016 (room 401): Loek Veenendaal

The Daniell integral

*Lecture 10:* 20 May 2016 (room 401): Willem van Zuijlen

Integration for functions with values in a partially ordered vector space

*Lecture 11:* 27 May 2016 (room 401): Willem Schouten

Measure and integration in locally compact spaces

*Lecture 12:* 3 June 2016 (room 401): David Kok

Fractals, semifractals and Markov operators

*Lecture 13:* 10 June 2016 (room 401): Erwin van der Meer

Differentiation on Euclidean space and functions of bounded variation