2 I. HECK E L-FUNCTIONS

The product of a finite number of locally compact topological spaces is again a

locally compact topological space. As a consequence of Tychonoff's theorem, one

has the following very useful result.

COROLLARY. Let {Xv}ves be a collection of locally compact topological spaces

with the Xv being compact spaces for all but a finite number of v, then the product

X — JJVeS Xv is locally compact.

We recall that a set G is called a topological group when G itself is a topological

space with the following properties:

(i) G is a group,

(ii) G is a Hausdorff space,

(iii) the inverse map G -^ G, g \-+ g~x is continuous,

(iv) the multiplication map G x C , (x, y) i— » xy is continuous.

As a topological group, G is a homogeneous space in the sense that for any two

elements g,h G G, there exists a homomorphism of G onto G which takes g into h;

a topology for G is completely determined by a system of open neighborhoods of

the identity element.

DEFINITIO N 2. A locally compact abelian group G is a topological group with

a commutative multiplication law.

A basic property of a locally compact topological group G is that any complex

valued, continuous, compactly supported function / on G is uniformly continuous

in the sense that given any e 0 there is some neighborhood of the identity, W

such that \f(g) - f(h)\ e for all

gh'1

G W.

A subgroup H of a topological group G, which is a topological space, is itself

a topological group. A topological subgroup H of G is called a closed subgroup if

as a subset of G, H is closed. Let H be a topological subgroup of G and consider

the coset space

G/H = {gH\g e G}.

Using the canonical mapping / : G — G/H, g i— gH, it is possible to introduce

a topology on G/H by declaring the family of subsets

W = {W CG/H :f-\W) is open in G}

to be the family of all open subsets. In this topology, the mapping / is clearly

continuous and when H is a closed normal subgroup, the quotient G/H is indeed a

topological subgroup. It is not difficult to show that if G is a compact group (resp.

locally compact) and N is a closed normal subgroup, then G/N is compact (resp.

locally compact).

Haar Measure

If X is a locally compact Hausdorff space, we denote by Y the smallest cr-ring

that contains the family of all compact subsets of X. The elements of Y are called

Borel sets and we suppose that X itself is a Borel set.

DEFINITIO N 3. A measure is a mapping from the ring of sets into the non-

negative, extended real numbers having the property that the image of the empty set