Definition of Haar Measure

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 $T_1$ group, we focus on locally compact Hausdorff group.

Let us quickly recall some preliminaries:

Locally compact Hausdorff Topological group:
- $(X,\tau)$ – Topological space ( $X$ is a set and $\tau$ is a collection of empty set, subsets of $X$ and the set $X$ itself such that $\tau$ is closed under arbitrary union and finite intersection.)
- $(X,\tau)$ – 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.
- $X$ is called a topology group, if it is a group and also a topological space along with the continuous group actions $X \times X \rightarrow X: (x_1,x_2)\rightarrow x_1x_2$ and $X\rightarrow X: x\rightarrow x^{-1}$
Borel regular measure:
- The $\sigma$ -algebra $\sigma(\Sigma)$ is a collection of subsets of $X$ that is closed under complement, countable union and intersection generated by $\Sigma\subset 2^X$ . Any set in $\sigma(\Sigma)$ is called a Borel set.
- A measure $\mu$ on a topological space $(X,\tau,\Sigma,\mu)$ with an underlying $\sigma$ -algebra is called Borel measure.
- A Borel measure that satisfies the following are called regular measure.
  -measure of any compact subset of $X$ is finite,
  -outer regular: $\mu(A)=inf\{\mu(U):A\subset U, U$ is open $\}\ \forall A\in \Sigma$
  -inner regular: $\mu(U)=sup\{\mu(K):K\subset U, K$ is compact $\}\$ for all open subsets $U$

Throughout, let $X$ be a locally compact Hausdorfff topological group (also a measure space). A non-zero measure $\mu$ on $X$ is called left Haar measure if it is

a Borel regular measure
a left invariant measure, that is, $\mu(gA)=\mu(A)$ for all $g\in X$ and all measurable sets $A$ .

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 $\mathbb{R}^n$ ; Lebesgue measure $\frac{1}{(2\pi)^n}d\theta_1\cdots d\theta_{n}$ on $S^1\times\cdots\times S^1$ , where $S^1$ denotes the circle group.

The following formula is one of the nice one that I cherish.

Suppose $X$ is homeomorphic to an open subset $U\subset \mathbb{R}^n$ in such a way that if we identify $X$ with $U$ , left translation is an affine map in the sense that $xy=A_xy+b_x$ , where $A_x$ is an invertible $n\times n$ matrix and $b_x$ is a vector in $\mathbb{R}^n$ . Then $|det\ A_x|^{-1}dx$ is a left Haar measure on $X$ .

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: $((0,\infty),\cdot)$
This is an open subset of the Euclidean line $\mathbb{R}$ .
Left translation: $xy=A_xy+b_x$ where $A_x=(x)$ and $b_x$ is zero.
Thus $x^{-1}dx$ is 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
$X=\bigg\{$ $\begin{pmatrix} x_1 & x_2 \\ 0 & 1 \end{pmatrix}$ , for all $x_1>0,x_2\in\mathbb{R}\bigg\}$
Identify each entry with $\begin{pmatrix} x_1\\x_2 \end{pmatrix}$ in $(0,\infty)\times\mathbb{R}$ (check that it is homeomorphism)
Left translation: given by the matrix $A_{x=(x_1,x_2)}:=$ $\begin{pmatrix} x_1 & 0 \\ 0 & x_1 \end{pmatrix}$ and the vector $b_x:=(0,x_2)$ since the matrix multiplication is given by
$\begin{pmatrix} x_1 & x_2\\ 0 & 1 \end{pmatrix}$ $*$ $\begin{pmatrix} y_1&y_2\\0 & 1 \end{pmatrix}$ $=$ $\begin{pmatrix} x_1y_1\\x_1y_2+x_2 \end{pmatrix}$ which is precisely given by $A_xy+b_x$ .
Hence $x_1^{-2}dx_1dx_2$ is the left Haar measure (and also the right Haar measure).However, the right translation is given by $yx=A_yx+b_y$ where $A_y=$ $\begin{pmatrix} y_1 & 0\\ y_2 & 1 \end{pmatrix}$ and $b_y$ is zero. Hence $x^{-1}dx_1dx_2$ is the right Haar measure.

In general, if $X$ is an open subset in $\mathbb{R}$ and we have $x=(x_1,x_2\cdots x_n)$ with $xy=F(x_1,\cdots,x_n,y_1,\cdots,y_n)$ where all the partial derivatives of $F$ exists and continuous, $\sigma_x(y)=xy$ and $\delta_{y}(x)=xy$ then $|J_{\sigma_x}|^{-1}dx$ is the left Haar measure and $|J_{\delta_x}|^{-1}dx$ is the right Haar measure where $|J_{\sigma_x}|$ denotes the modulus of the determinant of the Jacobian of the function $\sigma_x$ .

Special unitary group: $SU(2)$ (set of all $2\times 2$ unitary matrices with determinant 1.)
Homeomorphism between the sphere $S^3$ and the special unitary group $SU(2)$ :
$(x_1,x_2,x_3,x_4) \sim$ $\begin{pmatrix} x_1+ix_2 & -x_3+ix_4\\ x_3+ix_4 & x_1-ix_2 \end{pmatrix}$
where for $0\leq\theta\leq\pi,\ 0\leq\phi\leq \pi\ 0\leq\psi\leq 2\pi,$
$\begin{array}{rcl} x_1 & = & cos\ \theta \\ x_2 & = & sin\ \theta \ cos \ \phi\\ x_3 & = & sin\ \theta \ sin\ \phi\ cos\ \psi\\ x_4 & = & sin\ \theta \ sin\ \phi\ sin\ \psi \end{array}$
$\frac{1}{2\pi^2}sin^2\theta\ sin\phi\ d\theta d\phi d\psi$ is the left and right Haar measure.