Haar measure

Definition
Let $$G$$ be a locally compact group.

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:

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 $$\mu_r$$.

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.