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
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
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