HaarMeasure

Haar Measure on a Locally Compact Group

Let $G$ be a locally compact topological group. A right Haar measure $\lambda$ on $G$ is a non-zero Radon measure on $G$ that satisfies the following properties:

1. Translation-Invariance (Right-Invariance): For every $g \in G$ and every measurable set $A \subseteq G$ $$\lambda(A \cdot g) = \lambda(A),$$ where $A \cdot g = \lbrace ag : a \in A \rbrace$.

2. Non-Zero and Finite on Compact Sets: For any non-empty open set $U \subseteq G$ $$0 < \lambda(U) < \infty.$$

3. Regular: For every measurable set $A$ and every $\epsilon > 0$, there exist a compact set $K$ and an open set $U$ such that $$K \subseteq A \subseteq U,$$ and $$\lambda(U \setminus K) < \epsilon.$$

A left Haar measure satisfies similar properties but with left multiplication instead of right multiplication, i.e., $\lambda(g \cdot A) = \lambda(A)$.

Uniqueness

The Haar measure is unique up to multiplication by a positive scalar, meaning if $\mu$ is another right Haar measure, then there exists a positive constant $c$ such that $\mu = c \lambda$.

Examples

1. On $\mathbb{R}$, the Lebesgue measure is a Haar measure.
2. On $\mathbb{Z}$, the counting measure is a Haar measure.
3. On $\mathbb{R}^*$, the multiplicative group of non-zero real numbers, the measure $\lambda(dx) = \frac{dx}{|x|}$ is a Haar measure.