Haar measure

Definition

Left Haar measure

A left Haar measure is a left-translation-invariant countably additive regular nontrivial measure $\mu$ on the Borel subsets of $G$ . The conditions are explained below:

Condition name	Explanation
Borel subset	A subset in the $\sigma$ -algebra generated by open subsets of $G$
Left-translation-invariant measure	If $S\subseteq G$ is measurable, and $g\in G$ , then $gS$ is measurable, and $\mu (gS)=\mu (S)$ , where $gS=\{gs\mid g\in G\}$
Countably additive (part of the usual definition of measure)	If $S_{1},S_{2},\dots ,S_{n},\dots$ are all measurable sets that are pairwise disjoint, and their union is $S$ , then $\mu (S)$ is the sum of the values $\mu (S_{i})$ .
Regular Borel measure	$\mu (K)$ is finite for any compact subset $K$ . Also, for every Borel subset $S$ , $\mu (S)=\inf\{\mu (U),S\subseteq U,U\operatorname {open} \}$ and $\mu (S)=\sup\{\mu (UK),K\subseteq S,K\operatorname {compact} \}$
Nontrivial measure	$\mu (U)>0$ for any nonempty open subset $U$ of $G$

The left Haar measure for a locally compact group is unique up to scalar multiples, i.e., the quotient of any two left Haar measures is a scalar.

For a compact group, there is a unique choice of normalized Haar measure, i.e., a unique left Haar measure where the total measure of the group is $1$ .

Right Haar measure

A right Haar measure is a left-translation-invariant countably additive regular measure $\mu$ on the Borel subsets of $G$ .

The right Haar measure for a locally compact group is unique up to scalar multiples, i.e., the quotient of any two left Haar measures is a scalar.

Relationship between left and right Haar measure

Any left Haar measure on a group can be used to canonically define a right Haar measure: for a left Haar measure $\mu _{l}$ , define a correpsonding right Haar measure as:

$\mu _{r}(S):=\mu _{l}(S^{-1})$

where $S^{-1}=\{s^{-1}\mid s\in S\}$

This definition makes sense because $S$ is Borel if and only if $S^{-1}$ is. The left Haar measure conditions on $\mu _{l}$ give rise to the right Haar measures on $\mu _{r}$ .

Particular cases

Compact groups

For a compact group, the following additional features are true:

The left Haar measures coincide with the right Haar measures. In particular, we just talk of a Haar measure without talking of left or right. This Haar measure thus satisfies the additional condition that $\mu (S)=\mu (S^{-1})$ for any measurable subset $S$ .
Further, since the total measure of the group is finite, there is a natural choice of normalized Haar measure where the total measure of the group is $1$ .

Bi-invariant Haar measures?

A bi-invariant Haar measure is a measure that is both a left and a right Haar measure. Not all locally compact groups have bi-invariant Haar measures, but compact groups, locally compact abelian groups, and some other examples of locally compact groups have bi-invariant Haar measures. If a group has a bi-invariant Haar measure, then every left Haar measure is a right Haar measure, and hence a bi-invariant Haar measure.