A non-zero positive
measure
on the
-ring
of subsets
of a locally compact group
generated by the family of all compact subsets,
taking finite values on all compact subsets of
,
and satisfying either the
condition of left-invariance:
where
,
or the
condition of right-invariance:
where
.
Accordingly, one speaks of a
left-
or
right-invariant Haar measure.
Every Haar measure is
-regular,
that is,
for all
.
A left-invariant (and also a right-invariant) Haar measure exists and is unique,
up to a positive factor; this was established by
A. Haar
[1]
(under the additional assumption that the group
is separable).
If
is a continuous function of compact support on
,
then
is integrable relative to a left-invariant Haar measure on
,
and the corresponding integral is left-invariant (see
Invariant integration),
that is,
for all
.
A right-invariant Haar measure has the analogous property.
The Haar measure of the whole group
is finite if and only if
is compact.
If
is a left-invariant Haar measure on
,
then for every
the following equality holds:
where
is a continuous homomorphism of
into the multiplicative group
of positive real numbers that does not depend on the choice of the continuous function
of compact support on
.
The homomorphism
is called the
modulus
of
;
the measure
is a right-invariant Haar measure on
.
If
,
then
is called
unimodular;
in this case a left-invariant Haar measure is
also right-invariant and is called
(two-sided)
invariant.
In particular, every compact or discrete or Abelian
locally compact group, and also every connected semi-simple or
nilpotent Lie group, is unimodular. Unimodularity of a group
is also equivalent to the fact that every left-invariant Haar measure
on
is also
inversely invariant,
that is,
for all
.
If
is a
Lie group,
then the integral with respect to a left-invariant (right-invariant) Haar measure on
is defined by the formula
where the
are linearly independent left-invariant (right-invariant) differential
forms of order one on
(see
Maurer–Cartan form)
and
.
The
modulus of a Lie group
is defined by the formula
where
is the adjoint representation (cf.
Adjoint representation of a Lie group).
Examples.
1) The Haar measure on the additive group
and on the quotient group
(the group of rotations of the circle) is the same as the ordinary
Lebesgue measure.
2) The
general linear group
,
or
,
is unimodular, and the Haar measure has the form
where
for
and
for
,
and
is the Lebesgue measure in the Euclidean space of all matrices of order
over the field
.
If
is a locally compact group,
is a closed subgroup of it,
is the
homogeneous space
,
and
are the moduli of
and
,
respectively, and
is a continuous homomorphism of
into
whose restriction to
is given by the formula
then there exists a positive measure
on the
-ring
of sets
that is generated by the family of compact subsets of
;
it is uniquely determined by the condition:
where
is any continuous function of compact support on
,
,
and
for all continuous functions
of compact support on
.