# Measures on general topological groups

I am interested in the group algebras of non-locally compact groups. What references can you advise?

This is a wide question, so I list more concretely what I would like to see:

1. Here X can be even any Tychonoff space. Is there a description of the dual space of C0(X) as measures, or some close theorem? I have found a theorem in [H. König, Measure and integration] which describes measures as the dual of some space of semicontinuous functions. It is interesting, but I want the opposite: let the functions be "good", what measures will we get?

2. Are there any algebras, which consist more or less of measures, and can be associated to every topological group, or at lest to a Polish group? I've seen treatments of (left) uniformly continuous functions on groups and their dual space which is a Banach algebra. Is there a good survey of the topic? Are there other algebras?

Thank you, and obviously I would greet more than one answer.

EDIT: Question 1 is answered below, and for question 2 I have found meanwhile a paper and a bunch of references in it: Anthony To-Ming Lau, J. Ludwig: Fourier–Stieltjes algebra of a topological group, Advances in Mathematics 229, Issue 3 (2012), 2000–2023. For my purposes, this is enogh at the moment, so I close the question.

-
Just as a comment: Existence of quasi-invariant Radon measures, e.g. the Haar measure, for Polish group is equivalent to being locally compact. So any measure on a Polish group, which fails to be locally compact must vanish on a certain open set. Haar measure exist even without Hausdorffness, so that doesn't seem to be such an issue. –  Marc Palm Jan 22 '12 at 15:28
I don't seek for invariant measures... and indeed, I suppose that their supports may not contain open sets. That's the interest - to see what measures appear in this context. –  Yulia Kuznetsova Jan 23 '12 at 15:58

There are some results on the representation of certain functionals by measures in the paper

Smolyanov, O.G.; Fomin, S.V. Measures on linear topological spaces. Russ. Math. Surv. 31, No.4, 1-53 (1976); translation from Usp. Mat. Nauk 31, No.4(190), 3-56 (1976).

In fact, without local compactness the problem acquires completely new features, and it is instructive to consider the problems you are interested in for the case of a linear topological space. Then you will have to deal with completely new structures like spaces of cylindrical functions etc. There is some material of this kind in the above paper and in the later books by Daletsky and Fomin, Vakhania et al, etc.

-
Thank you, it definitely contains answers to some of my questions. Indeed, the first example of $X$ under question is a Banach space considered as an additive group. –  Yulia Kuznetsova Jan 20 '12 at 22:46
At the first glance it seems that my second question (convolutions of measures) is not treated there. –  Yulia Kuznetsova Jan 20 '12 at 22:50

V. S. Varadarajan, MEASURES ON TOPOLOGICAL SPACES, AMS Transl. 48 (1965) 161--228.

In fact, much of what I was remembering when I wrote my previous reply is actually here. measures on topological spaces as dual to continuous functions on the space, or to bounded continuous functions on the space. (Also, beware of an error in the appendix.)

-
Thank you! This is really close to what I seek. –  Yulia Kuznetsova Jan 23 '12 at 16:01

Since a topological group has a natural uniform structure, the appropriate framework is, I think, the space of uniform measures. This was studied by the Czechoslovakian school. In functional analytical terms, it is the dual of the space of bounded, uniformly continuous functions on a uniform space---not with the uniform norm, of course (that would produce measures on the Samuel compactification). The rather more elaborate structure used is described in the article "Measures as functionals on uniformly continuous functions" by Pachl. A strong hint that this is the right concept is provided by the fact that the resulting space of measures has the natural universal property for bounded, uniformly continuous functions from the uniform space into complete, locally convex spaces. Pachl has also considered measures on topological groups in this context---see his papers "Uniform measures on topological groups" and "Uniform measures and convolutions on topological groups"---all readily available on internet.

-

A good book for measures on topological spaces as dual to continuous functions on the space, or to bounded continuous functions on the space ... Gillman & Jerison, RINGS OF CONTINUOUS FUNCTIONS

-
The book is interesting indeed; but could you please specify which place of it is relevant to my question? I couldn't find a description of the dual space of functions, and it seems like measures appear there only in the context of measurable cardinals. –  Yulia Kuznetsova Jan 20 '12 at 16:43
You are right. What I was remembering is not in this book! I will post another reply. –  Gerald Edgar Jan 21 '12 at 2:23