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HaarMeasure
Let
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Translation-Invariance (Right-Invariance): For every
$g \in G$ and every measurable set$A \subseteq G$ $$\lambda(A \cdot g) = \lambda(A),$$ where$A \cdot g = \lbrace ag : a \in A \rbrace$ . -
Non-Zero and Finite on Compact Sets: For any non-empty open set
$U \subseteq G$ $$0 < \lambda(U) < \infty.$$ -
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.,
The Haar measure is unique up to multiplication by a positive scalar, meaning if
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$\mathbb{R}$ , the Lebesgue measure is a Haar measure. - On
$\mathbb{Z}$ , the counting measure is a Haar measure. - On
$\mathbb{R}^*$ , the multiplicative group of non-zero real numbers, the measure$\lambda(dx) = \frac{dx}{|x|}$ is a Haar measure.