Historical notes

Abstract topological groups were first defined by Schreier in 1926, though the idea was implicit in much earlier work on continuous groups of transformations. The subject has its origins in Klein's programme (1872) to study geometries through the transformation groups associated with them, and in Lie's theory of continuous groups arising from the solution of differential equations. The ‘classical groups’ of geometry (general linear groups, unitary groups, symplectic groups, etc.) are in fact Lie groups, that is, they are analytic manifolds and their group operations are analytic functions. On the other hand, Killing and Cartan showed (1890) that all simple Lie groups are classical groups, apart from a finite number of exceptional groups.

In 1900 Hilbert posed the problem (No. 5 of his famous list) whether every continuous group of transformations of a finite-dimensional real or complex space is a Lie group. The twentieth-century habit of axiomatising everything led to a more abstract formulation of this problem. A topological group is a topological space which is a group with continuous group operations, and the question is: What topological conditions on a topological group will ensure that it has an analytic structure which makes it a Lie group? Since integration was a major tool in the study of Lie groups, especially their representations, it became important to establish the existence of appropriate integrals on general classes of topological groups.

In 1969, and again in 1972, I gave an introductory course on topological groups at the University of London. The audience consisted largely of first-year postgraduate students in algebra or number theory and the course was designed to meet their needs, which I took to be a knowledge of basic facts about various general types of topological groups and enough about the Haar integral to enable them to appreciate its use in such contexts as representation theory and algebraic number theory. Some of the students had little or no background in topology and for this reason topological concepts were developed ab initio, whereas the rudiments of group theory were assumed.

After the 1969 course there was some demand from new students for copies of the notes and so it was decided to produce a duplicated set of notes of the 1972 lectures. I am extremely grateful to Robert Coates, Michael Rutter and Anthony Solomonides for all their hard work in preparing the mimeographed version on which the present notes are based. I have made a number of minor changes and amplified some passages when this seemed advisable for a wider audience. I have also added an informal section on the representation theory of compact groups which I intended to include in the course but for which there was no time. Its purpose is to illustrate one practical application of the Haar integral and to encourage further reading.

Abstract integrals

We seek to construct an ‘integral’ on a locally compact group. This may be done by a variety of methods; we wish to avoid measure-theoretic methods and so we take as our model the construction of the Riemann integral on (R, +). There we have the set R of all Riemannintegrable functions (vanishing outside some interval), and a map ∫ : R → R defined by

which satisfies:

1. (i) Linearity: ∫(λf + μg) = μ∫g.

2. (ii) Positivity: f ≥ 0 ⇒ ∫f ≥ 0.

3. (iii) Translation-invariance: if for some fixed a ∈ R and all x ∈ R, g(x) = f (x + a) then

∫g = ∫f.

These three properties actually characterize the Riemann integral up to a scalar multiple. (Exercise.) The same is true if we consider only continuous functions on R, and this suggests that we should try to construct, for more general groups, an integral with similar properties for realvalued continuous functions.

From now on we shall only consider locally compact spaces and groups, and we shall always assume that they are Hausdorff.

For a space X, K(X) denotes the set of all continuous functions X → R with compact support, where

support(f) = closure of {x ∈ X;f(x) 0}.