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 $f\in C_c\left(G\right),$ $ℂ\text{-valued}$ functions on $G$ with compact support, which satisfies:

H1:	${\displaystyle {\int }_G}-dg:C_c\left(G\right)\to ℂ$ is a continuous linear functional.
H2:	(Positivity) ${\displaystyle {\int }_G}f\left(g\right)dg\ge 0$ for all $f$ taking non-negative real values. There is such an $f$ whose integral is non-zero.
H3:	(Left Invariance) ${\displaystyle {\int }_G}f\left(hg\right)dg={\displaystyle {\int }_G}f\left(g\right)dg$ for all integrable functions $f$ and all $h\in G\text{.}$

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

The conditions H1 and H2 are related to the topology on $G\text{.}$

A $ℂ\text{-valued}$ 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 $\{g\in G:f\left(g\right)\ne 0\}$ is compact. The set $C_c\left(G\right)$ of such functions form a subspace of all $ℂ\text{-valued}$ functions on $G\text{.}$ Such a function will be called positive if it is non-trivial and takes values in $[0,\infty )\text{.}$ A linear functional on $C_c\left(G\right)$ is a linear map from $C_c\left(G\right)$ to $ℂ\text{.}$ A positive linear functional on $C_c\left(G\right)$ is is non-trivial linear functional that takes values in $[0,\infty )$ for all positive $f\text{.}$ We consider positive linear functionals which are continuous with respect to the topology on $C_c\left(g\right)$ given by the sup-norm, ${\Vert f\Vert }_{\infty }=\text{sup}\{\mid f\left(g\right)\mid :g\in G\}\text{.}$ (Setting $d(f_1,f_2)={\Vert f_1-f_2\Vert }_{\infty }$ gives a metric on $C_c\left(G\right)\text{).}$

Measure theory shows every positive continuous linear functional $\mu :C_c\left(G\right)\to ℂ$ is given by integration with respect to a unique regular Borel measure $\mu \text{.}$ (A Borel measure is a measure on the $\sigma \text{-algebra}$ $ℳ$ generated by the open subsets of $G\text{.}$ A Borel measure is called regular if compact sets have finite measure and for every $E$ of finite measure $\mu \left(E\right)=\text{sup}\{\mu \left(K\right):K\subseteq E\}\text{.}$ Given a Borel measure $\mu ,$ $L^1(G,ℂ)$ denotes the space of $ℂ\text{-valued}$ functions which are absolutely integral with respect to the measure. For $f\in L^1(G,ℂ)$ we denote its integral by $\mu \left(f\right)={\int }_{G}f\left(g\right)d\mu \left(g\right)\text{.}$ When the measure is regular $C_c\left(G\right)$ is a linear subspace of $L^1(G,ℂ)\text{.}$ Then $\mu \left(f\right)={\int }_{G}f\left(g\right)d\mu \left(g\right)$ defines a continuous positive linear functional on $C_c\left(G\right)\text{.}$ This sets up a one to one correspondence between regular Borel measures and positive continuous linear functionals on $C_c\left(G\right)\text{.}$

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

Left $G$ action, ${}^{h}f$ on functions $f$ with domain $G,$ is defined by setting ${}^{h}f\left(g\right)=f\left(h^{-1}g\right)$ and right $G\text{-action}$ is defined by $f^h\left(g\right)=f\left(g{h}^{-1}\right)\text{.}$ Because $G$ is a topological group, if $X\subseteq G$ is open, then so too are both $hX$ and $Xh$ for every $h\in G\text{.}$ Similarly for compact subsets and similarly too for all sets in $ℳ\text{.}$

We deduce firstly that if $f$ is continuous of compact support then so too are both ${}^{h}f$ and $f^h$ for any $h\in G\text{.}$ Secondly we have actions on Borel measures taking regular measures to regular measures. Given a measure $d\mu \left(g\right)$ and $h\in G$ we form measures $d\mu \left(hg\right)$ and $d\mu \left(gh\right),$ the first giving $X\in ℳ$ measure $\mu \left(hX\right)$ and the second giving $X$ measure $\mu \left(Xh\right)\text{.}$ Thus it makes sense to change variables in integrals as follows $$\begin{array}{rcl}{\displaystyle {\int }_G{}^{h}f\left(g\right)d\mu \left(g\right)}& =& {\displaystyle {\int }_{g\in G}f\left(h^{-1}g\right)d\mu \left(g\right)}\\ & =& {\displaystyle {\int }_{hg\in G}f\left(h^{-1}hg\right)d\mu \left(hg\right)}\\ & =& {\displaystyle {\int }_{hg\in G}f\left(g\right)d\mu \left(hg\right)}\\ & =& {\displaystyle {\int }_{g\in G}f\left(g\right)d\mu \left(hg\right),}\end{array}$$ and similarly for right $G\text{-action.}$ Thus we have $${\int }_G{}^{h}f\left(g\right)d\mu \left(g\right)={\int }_{G}f\left(g\right)d\mu \left(hg\right)\text{and}{\int }_G{f}^h\left(g\right)d\mu \left(g\right)={\int }_{G}d\left(g\right)d\mu \left(gh\right)\text{.}$$

Exercise: If we define left $G\text{-action}$ on measures by ${}^h\mu \left(X\right)=\left(h^{-1}X\right)$ and right $G\text{-action}$ on measures by $\mu {\left(X\right)}^h=\mu \left(X{h}^{-1}\right)$ then for $f\in C_c\left(G\right),$ $${}^h\mu \left(f\right)=\mu \left({}^{h^{-1}}f\right)\text{and}{\mu }^h\left(f\right)=\mu \left(f^{h^{-1}}\right)\text{.}$$

A Borel measure on $G$ is called left invariant if: $$\mu \left(hX\right)=\mu \left(X\right)\text{for all}\hspace{0.17em}h\in G\hspace{0.17em}\text{and}\hspace{0.17em}X\in ℳ\text{.}$$ (This is equivalent to ${}^h\mu =\mu$ for all $h\in G\text{.)}$ 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 $\mu$ we write $d\mu \left(g\right)=dg\text{.}$ We can express left invariance thus: $${\int }_G{}^{h}f\left(g\right)dg={\int }_{G}f\left(g\right)dg\text{for all}\hspace{0.17em}h\in G\text{.}$$ This is equivalent, to H3 above. In terms of change of variable this becomes simply, $$d\left(hg\right)=dg\text{for all}\hspace{0.17em}h\in G\text{.}$$

Example: Discrete Groups.

Every group $G$ is locally compact, with respect to the discrete topology. Counting measure on subsets, $\mu \left(X\right)=\text{Card}\left(X\right)$ for $X\subseteq G,$ is a left Haar measure on $G\text{.}$

The compact subsets are the finite subsets of $G\text{.}$ For $f\in C_c\left(G\right),$ $f\left(g\right)=0$ for all but finitely many $g$ and $${\int }_{G}f\left(g\right)dg=\sum _{g\in G}f\left(g\right)\text{.}$$

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 $\mu$ is a left Haar measure on $G$ then given any $h\in G$ so too is $\mu \prime$ where $\mu \prime \left(X\right)=\mu \left(Xh\right)$ for $X\in ℳ\text{.}$ Hence these measures differ by a positive real multiple, which depends on $h\text{.}$ This multiple does depend on not the choice of left Haar measure $\mu \text{.}$

The function $\mathrm{\Delta }:G\to (0,\infty ),$ defined by $$\mu \left(Xh\right)=\mathrm{\Delta }\left(h\right)\mu \left(X\right),\text{for all}\hspace{0.17em}X\in ℳ,$$ is called the modular function of $G\text{.}$

We can express this alternatively as $$d\left(gh\right)=\mathrm{\Delta }\left(h\right)dg\text{.}$$ Integration gives: $$\begin{array}{rcl}{\displaystyle {\int }_G{f}^h\left(g\right)dg}& =& {\displaystyle {\int }_{G}f\left(g\right)d\left(gh\right)}\\ & =& {\displaystyle {\int }_{G}f\left(g\right)\mathrm{\Delta }\left(h\right)dg}\\ & =& {\displaystyle \mathrm{\Delta }\left(h\right){\int }_{G}f\left(g\right)dg\text{.}}\end{array}$$ Thus the modular function $\mathrm{\Delta }$ can alternatively be defined by $${\int }_G{f}^h\left(g\right)dg=\mathrm{\Delta }\left(h\right){\int }_{G}f\left(g\right)dg\text{for all}\hspace{0.17em}f\in C_c\left(G\right)\text{.}$$

The modular function of $G$ is a continuous homomorphism from $G$ to $(0,\infty )\text{.}$

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. $\square$

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 ${\mathrm{\Delta }}^{-1}\left(g\right)dg\text{.}$

		Proof.
	$${\mathrm{\Delta }}^{-1}\left(gh\right)d\left(gh\right)={\mathrm{\Delta }}^{-1}\left(g\right){\mathrm{\Delta }}^{-1}\left(h\right)d\left(gh\right)={\mathrm{\Delta }}^{-1}\left(g\right){\mathrm{\Delta }}^{-1}\left(h\right)\mathrm{\Delta }\left(h\right)dg={\mathrm{\Delta }}^{-1}\left(g\right)dg\text{.}$$ Note this argument is entirely algebraic. $\square$

There is a second way to relate left and right Haar measure. If $dg$ is a left Haar measure then $d\left(g^{-1}\right)$ is a right Haar measure. In terms of integrals $d\left(g^{-1}\right)$ is defined by, $${\int }_{G}f\left(g\right)d\left(g^{-1}\right)≔{\int }_{G}f\left(g^{-1}\right)dg\text{.}$$ In terms of measures, if $dg=d\mu \left(g\right)$ then $d\left(g^{-1}\right)=d{\mu }^{*}\left(g\right),$ where ${\mu }^{*}\left(X\right)=\mu \left(X-1\right),$ $X^{-1}≔\{g^{-1}:g\in X\}\text{.}$ Since $d\left(g^{-1}\right)$ and ${\mathrm{\Delta }}^{-1}\left(g\right)dg$ are right Haar measures they must differ by a positive multiple. In fact they are always equal and we have $${\int }_{G}f\left(g^{-1}\right)dg={\int }_{G}f\left(g\right){\mathrm{\Delta }}^{-1}\left(g\right)dg\text{.}$$

In general ${\displaystyle {\int }_G}f\left(g^{-1}\right)dg$ is not equal to ${\displaystyle {\int }_G}f\left(g\right)dg$ unless $G$ is unimodular.

Other Examples of Haar measures:

(1)	$ℝ$ under addition. This has Haar measure $dx,$ giving the usual Riemann integral $\mu \left(f\right)={\displaystyle {\int }_a^b}f\left(x\right)dx$ for $f$ a continuous function zero outside the closed bounded interval $[a,b]\text{.}$
(2)	The group, $\{(0,\infty ),\times \},$ of positive real numbers under multiplication. This has Haar measure $dx/x\text{.}$
(3)	${GL}_n\left(ℝ\right),$ invertible $n\times n$ matrices $\left(x_{ij}\right)$ with entries $x_{ij}\in ℝ\text{.}$ This has Haar measure $\prod _{i,j}d{x}_{ij}/{\mid \text{det}\left(x_{ij}\right)\mid }^n\text{.}$ This group is unimodular.
(4)	The group of upper triangular $n\times n\text{-matrices}$ in ${GL}_n\left(ℝ\right)\text{.}$ This has left Haar measure $\prod _{i<j}d{x}_{ij}/\prod _i{\mid x_{ii}\mid }^i\text{.}$ This group is not unimodular unless $n=1\text{.}$

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