Integration of functions on a
topological group
that has a certain invariant property with respect to the group operations. Thus, let
be a locally compact topological group, let
be the vector space of all continuous complex-valued functions with compact support on
and let
be an
integral
on
,
that is, a positive
linear functional
on
(
for
).
The integral
is called
left-invariant
(or
right-invariant)
if
(or
)
for all
,
;
here
The integral
is called
(two-sided)
invariant
if it is both left- and right-invariant. The mapping
,
where
,
,
defines a one-to-one correspondence between the classes
of left- and right-invariant integrals on
.
If
,
then
is called
inversion invariant.
There exists on every locally compact group
a non-zero left-invariant integral; it is unique up to a numerical factor (the
Haar–von Neumann–Weil theorem).
This integral is called the
left Haar integral.
The following equation holds:
where
,
and
is a continuous homomorphism from the group
into the multiplicative group of positive real numbers (a
positive character).
Furthermore,
.
The character
is called the
modulus
of
.
If
,
then
is called
unimodular.
In this case
is a two-sided invariant integral.
In particular, every compact group (where
,
)
and every discrete group (where
,
)
is unimodular.
According to the
Riesz theorem,
every integral on
is a
Lebesgue integral
with respect to some
Borel measure
which is uniquely defined in the class of Borel
measures that are finite on each compact subset
.
The left- (or right-) invariant measure
corresponding to the left (right) Haar integral on
is called the left (right)
Haar measure
on
.
Let
be a closed subgroup of
and let
be the modulus of
.
If
can be extended to a continuous positive character of
(cf.
Character of a group),
then there exists on the left homogeneous space
a
relatively invariant integral
,
that is, a positive functional on the space
of continuous functions with compact support on
that satisfies the identity
for all
,
;
here
and
is the modulus of
.
This integral is defined by the rule
,
where
is the left Haar integral on
and
is a function on
such that
(
is the left Haar integral on
and
is the restriction of
to
.)
This is well-defined since
is a mapping from
onto
and
when
.
The notion of an invariant mean (cf.
Invariant average)
is closely related to that of invariant integration.