1. ABSTRAC T HARMONIC ANALYSIS: A SURVEY 3

under this mapping is 0 and the measure of any countable, disjoint union of sets,

which is itself a member of the ring is the sum of the measures of the individual

sets. A Borel measure on Y is a mapping \i, which is a measure on Y and the

measure of any compact subset of Y is finite.

DEFINITIO N 4. A right Haar measure on a locally compact topological group

G is a Borel measure such that

(i) the measure of any non-empty open set is positive, and

(ii) n(Eg) = 11(E), E€Y,gGG.

A left Haar measure is defined similarly by replacing (ii) by the condition

li(gE) = »(E).

In this chapter we shall be mainly interested in locally compact abelian groups

and we will refer to a right (resp. left) Haar measure simply as a Haar measure.

The main result about Haar measures is the following.

T H E O R E M 2. (Weil) If G is a locally compact group, then it admits a right Haar

measure uniquely determined up to a constant factor.

The existence of a Haar measure on a locally compact abelian group is equiv-

alent to the existence of an invariant positive functional on the space CC(G) of all

continuous compactly supported, complex valued functions on G.

DEFINITIO N 5. Let G be a locally compact abelian group. A relative invariant

measure dfi on G with multiplier \ is a measure which satisfies the relation

dfi(gx) = x(g)d^{x), g eG.

T H E O R E M 3. A relative invariant measure on a locally compact abelian group

is unique up to a constant factor.

This is a well-known fact whose proof depends on the uniqueness of Haar mea-

sure. For a proof see Weil ([209], Chapter II, p.40).

DEFINITIO N 6. If G is a locally compact group and dfjb is a Haar measure,

L\{G) = Li(G,d/jb) is the linear space of all measurable functions on G.

Duality Theory

Let T = {z E C : z - z = 1} be the circle group with the topology it inherits as

a subset of the complex numbers C. Let G be a locally compact abelian group and

let

G = Homc(G,T),

be the set of all continuous homomorphisms from G to T; it is in an obvious way

a topological group. On G we introduce a topology by decreeing that convergence

in G is equivalent to uniform convergence on compact subsets of G (the elements

of G being functions on G). In fact a basis for a system of neighborhoods of the