Topological Groups and Haar Measure

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 27 November 2014

Topological Groups and Haar Measure

Let G be a topological group, that is a group with a topology in which the operations multiplication and taking inverse are continuous.

We assume further G is locally compact Hausdorff with respect to its topology.

A locally compact Hausdorff topological group has a left Ilaar measure. Left Haar measure is unique up to multiplication by a real positive constant.

Giving a left Haar measure is equivalent to giving an integral on functions fCc(G), -valued functions on G with compact support, which satisfies:

H1: G-dg:Cc(G) is a continuous linear functional.
H2: (Positivity)
Gf(g)dg0 for all f taking non-negative real values.
There is such an f whose integral is non-zero.
H3: (Left Invariance)
Gf(hg)dg=Gf(g)dg for all integrable functions f and all hG.

The three conditions above characterize left Haar integral up to a positive multiple.

The conditions H1 and H2 are related to the topology on G.

A -valued function on G is said to have compact support if it zero on the complement of a compact subset of G, or equivalently the closure of {gG:f(g)0} is compact. The set Cc(G) of such functions form a subspace of all -valued functions on G. Such a function will be called positive if it is non-trivial and takes values in [0,). A linear functional on Cc(G) is a linear map from Cc(G) to . A positive linear functional on Cc(G) is is non-trivial linear functional that takes values in [0,) for all positive f. We consider positive linear functionals which are continuous with respect to the topology on Cc(g) given by the sup-norm, f=sup{f(g):gG}. (Setting d(f1,f2)=f1-f2 gives a metric on Cc(G)).

Measure theory shows every positive continuous linear functional μ:Cc(G) is given by integration with respect to a unique regular Borel measure μ. (A Borel measure is a measure on the σ-algebra generated by the open subsets of G. A Borel measure is called regular if compact sets have finite measure and for every E of finite measure μ(E)=sup{μ(K):KE}. Given a Borel measure μ, L1(G,) denotes the space of -valued functions which are absolutely integral with respect to the measure. For fL1(G,) we denote its integral by μ(f)=Gf(g)dμ(g). When the measure is regular Cc(G) is a linear subspace of L1(G,). Then μ(f)=Gf(g)dμ(g) defines a continuous positive linear functional on Cc(G). This sets up a one to one correspondence between regular Borel measures and positive continuous linear functionals on Cc(G).

The one to one correspondence between measure and positive continuous linear functionals arises from the locally compact topology on G. We now consider the group action.

Left G action, hf on functions f with domain G, is defined by setting hf(g)=f(h-1g) and right G-action is defined by fh(g)=f(gh-1). Because G is a topological group, if XG is open, then so too are both hX and Xh for every hG. Similarly for compact subsets and similarly too for all sets in .

We deduce firstly that if f is continuous of compact support then so too are both hf and fh for any hG. Secondly we have actions on Borel measures taking regular measures to regular measures. Given a measure dμ(g) and hG we form measures dμ(hg) and dμ(gh), the first giving X measure μ(hX) and the second giving X measure μ(Xh). Thus it makes sense to change variables in integrals as follows Ghf(g) dμ(g) = gGf (h-1g)dμ (g) = hgGf (h-1hg) dμ(hg) = hgG f(g)dμ (hg) = gGf(g) dμ(hg), and similarly for right G-action. Thus we have Ghf(g) dμ(g)= Gf(g)dμ(hg) andGfh (g)dμ(g)= Gd(g)dμ(gh).

Exercise: If we define left G-action on measures by hμ(X)=(h-1X) and right G-action on measures by μ(X)h=μ(Xh-1) then for fCc(G), hμ(f)=μ (h-1f) andμh(f) =μ(fh-1).

A Borel measure on G is called left invariant if: μ(hX)=μ(X) for allhGand X. (This is equivalent to hμ=μ for all hG.) A left invariant regular Borel measure is called a left Haar measure.

(1) Left Haar measure is unique up to multiplication by a positive constant.
(2) The Haar measure of compact subsets K of G with open interior is non-zero.
(3) A Haar measure is determined by fixing the measure of any compact subset K of G with open interior.

Having made a fixed a choice of left Haar measure μ we write dμ(g)=dg. We can express left invariance thus: Ghf(g) dg=Gf(g)dg for allhG. This is equivalent, to H3 above. In terms of change of variable this becomes simply, d(hg)=dg for allhG.

Example: Discrete Groups.

Every group G is locally compact, with respect to the discrete topology. Counting measure on subsets, μ(X)=Card(X) for XG, is a left Haar measure on G.

The compact subsets are the finite subsets of G. For fCc(G), f(g)=0 for all but finitely many g and Gf(g)dg= gGf(g).

A locally compact topological group also has a right Haar measures unique up to positive multiple. In general a left Haar measure is not a right Haar measure. The relationship between left Haar measure and right Haar measure is described by the modular function of G which is defined below.

If μ is a left Haar measure on G then given any hG so too is μ where μ(X)=μ(Xh) for X. Hence these measures differ by a positive real multiple, which depends on h. This multiple does depend on not the choice of left Haar measure μ.

The function Δ:G(0,), defined by μ(Xh)=Δ(h) μ(X),for allX, is called the modular function of G.

We can express this alternatively as d(gh)=Δ (h)dg. Integration gives: Gfh(g)dg = Gf(g) d(gh) = Gf(g) Δ(h)dg = Δ(h)G f(g)dg. Thus the modular function Δ can alternatively be defined by Gfh(g)dg= Δ(h)G f(g)dgfor all fCc(G).

The modular function of G is a continuous homomorphism from G to (0,).

Exercise: Show the modular function is well defined and a homomorphism.

A Left Haar measure on a locally compact topological group G is a right Haar measure if and only if the modular function of G is trivial.

Proof.

This follows directly from the definition of the modular function.

A locally compact topological group for which left Haar measures are right Haar measures, i.e. those whose modular function is trivial, are called unimodular.

Exercise/Examples

(1) Groups G with the discrete topology are unimodular.
(2) If G is locally compact abelian then it is unimodular.
(3) If G is compact then it is unimodular.
(Hint: the image of a compact set under a continuous map is compact)

If a left Haar measure on G is given by dg then a tight Haar measure is given by Δ-1(g)dg.

Proof.

Δ-1(gh) d(gh)=Δ-1 (g)Δ-1(h) d(gh)=Δ-1 (g)Δ-1(h) Δ(h)dg= Δ-1(g)dg.

Note this argument is entirely algebraic.

There is a second way to relate left and right Haar measure. If dg is a left Haar measure then d(g-1) is a right Haar measure. In terms of integrals d(g-1) is defined by, Gf(g)d (g-1) Gf(g-1)dg. In terms of measures, if dg=dμ(g) then d(g-1)=dμ*(g), where μ*(X)=μ(X-1), X-1{g-1:gX}. Since d(g-1) and Δ-1(g)dg are right Haar measures they must differ by a positive multiple. In fact they are always equal and we have Gf(g-1)dg= Gf(g)Δ-1 (g)dg.

In general Gf(g-1)dg is not equal to Gf(g)dg unless G is unimodular.

Other Examples of Haar measures:

(1) under addition. This has Haar measure dx, giving the usual Riemann integral μ(f)=abf(x)dx for f a continuous function zero outside the closed bounded interval [a,b].
(2) The group, {(0,),×}, of positive real numbers under multiplication. This has Haar measure dx/x.
(3) GLn(), invertible n×n matrices (xij) with entries xij. This has Haar measure i,jdxij/det(xij)n. This group is unimodular.
(4) The group of upper triangular n×n-matrices in GLn(). This has left Haar measure i<jdxij/ixiii. This group is not unimodular unless n=1.

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