The name “Haar measure” came into existence after Alfred Haar in 1933 introduced invariant measures (invariant with respect to the group operation) on topological groups. Although Haar measure can be defined on locally compact group, we focus on locally compact Hausdorff group.
Let us quickly recall some preliminaries:
- Locally compact Hausdorff Topological group:
– Topological space (
is a set and
is a collection of empty set, subsets of
and the set
itself such that
is closed under arbitrary union and finite intersection.)
– Hausdorff topological space if the underlying topology is Hausdorff, that is, any two distinct points have disjoint neighborhood.
Most of the spaces I consider in practice are Hausdorff spaces. Simple example would be Euclidean space. One of the simplest counter example is cofinite topology and one of the other rich counter example is Pseudometric spaces.is called a topology group, if it is a group and also a topological space along with the continuous group actions
and
- Borel regular measure:
- The
-algebra
is a collection of subsets of
that is closed under complement, countable union and intersection generated by
. Any set in
is called a Borel set.
- A measure
on a topological space
with an underlying
-algebra is called Borel measure.
- A Borel measure that satisfies the following are called regular measure.
-measure of any compact subset ofis finite,
-outer regular:is open
-inner regular:is compact
for all open subsets
- The
Throughout, let be a locally compact Hausdorfff topological group (also a measure space). A non-zero measure
on
is called left Haar measure if it is
- a Borel regular measure
- a left invariant measure, that is,
for all
and all measurable sets
.
Similarly we define right Haar measure. The construction of Haar measure and proving its uniqueness are beautiful, which we will save it for later. In this note, let us consider only examples.
Some trivial examples are cardinality acts as Haar measure on finite groups (can be normalized) and for infinite discrete groups; the usual Lebesgue measure on ; Lebesgue measure
on
, where
denotes the circle group.
The following formula is one of the nice one that I cherish.
Suppose is homeomorphic to an open subset
in such a way that if we identify
with
, left translation is an affine map in the sense that
, where
is an invertible
matrix and
is a vector in
. Then
is a left Haar measure on
.
Similarly we can state it for right Haar measure with the appropriate right translation.
Let us recall at first a few examples applying the above theorem.
Examples:
- Multiplicative Group:
This is an open subset of the Euclidean line.
Left translation:where
and
is zero.
Thusis the left Haar measure on the multiplicative group.
In fact, it is the right Haar measure as well. - The upper half plane or the set
, for all
Identify each entry within
(check that it is homeomorphism)
Left translation: given by the matrixand the vector
since the matrix multiplication is given by
which is precisely given by
.
Henceis the left Haar measure (and also the right Haar measure).However, the right translation is given by
where
and
is zero. Hence
is the right Haar measure.
In general, if is an open subset in
and we have
with
where all the partial derivatives of
exists and continuous,
and
then
is the left Haar measure and
is the right Haar measure where
denotes the modulus of the determinant of the Jacobian of the function
.
- Special unitary group:
(set of all
unitary matrices with determinant 1.)
Homeomorphism between the sphereand the special unitary group
:
where for
is the left and right Haar measure.
References:
S. C. Bagchi, S. Madan, A. Sitaram and U. B. Tiwari, A first course on Representation theory and linear Lie groups, Universities Press, Hyderabad (2000)