# Haar measure

A non-zero positive measure $\mu$ on the $\sigma$-ring $M$ of subsets $E$ of a locally compact group $G$ generated by the family of all compact subsets, taking finite values on all compact subsets of $G$, and satisfying either the condition of left-invariance: $$\forall E \in M, ~ \forall g \in G: \qquad \mu(E) = \mu(g E),$$ where $g E = \{ g x \in G \mid x \in E \}$, or the condition of right-invariance: $$\forall E \in M, ~ \forall g \in G: \qquad \mu(E) = \mu(E g),$$ where $E g = \{ x g \in G \mid x \in E \}$. Accordingly, one speaks of a left- or right-invariant Haar measure. Every Haar measure is $\mu$-regular, i.e., $$\forall E \in M: \qquad \mu(E) = \sup(\{ \mu(K) \in \mathbf{R}_{\geq 0} \mid K \subseteq E ~ \text{and} ~ K ~ \text{is a compactum} \}).$$

A left-invariant (and also a right-invariant) Haar measure exists and is unique, up to a positive factor; this was established by A. Haar () (under the additional assumption that the group $G$ is separable).

If $f \in {C_{c}}(G)$, then $f$ is integrable relative to a left-invariant Haar measure on $G$, and the corresponding integral is left-invariant, i.e., $$\forall g_{0} \in G: \qquad \int_{G} f(g) ~ \mathrm{d}{\mu(g)} = \int_{G} f(g_{0} g) ~ \mathrm{d}{\mu(g)}.$$ A right-invariant Haar measure has the analogous property. The Haar measure of the whole group $G$ is finite if and only if $G$ is compact.

If $\mu$ is a left-invariant Haar measure on $G$, then the following equality holds: $$\forall f \in {C_{c}}(G), ~ \forall g_{0} \in G: \qquad \int_{G} f(g g_{0}^{-1}) ~ \mathrm{d}{\mu(g)} = \Delta(g_{0}) \int_{G} f(g) ~ \mathrm{d}{\mu(g)},$$ where $\Delta$ is a continuous homomorphism of $G$ into the multiplicative group $\mathbf{R}^{+}$ of positive real numbers that does not depend on the choice of $f$. The homomorphism $\Delta$ is called the modular function of $G$; the measure $\Delta(g^{-1}) ~ \mathrm{d}{\mu(g)}$ is a right-invariant Haar measure on $G$. If $\Delta(g) = 1$ for all $g \in G$, then $G$ is called unimodular; in this case a left-invariant Haar measure is also right-invariant and is called (two-sided) invariant. In particular, every compact or discrete or Abelian locally compact group, and also every connected semi-simple or nilpotent Lie group, is unimodular. Unimodularity of a group $G$ is also equivalent to the fact that every left-invariant Haar measure $\mu$ on $G$ is also inversely invariant, i.e., $\mu(E^{-1}) = \mu(E)$ for all $E \in M$.

If $G$ is a Lie group, then the integral with respect to a left-invariant (right-invariant) Haar measure on $G$ is defined by the formula $$\int_{G} f(x) ~ \mathrm{d}{\mu(x)} = \int_{G} f ~ \omega_{1} \wedge \cdots \wedge \omega_{n},$$ where the $\omega_{i}$’s are linearly independent left-invariant (right-invariant) differential forms of order $1$ on $G$ (see the Maurer–Cartan form) and $n = \dim(G)$. The modular function of a Lie group $G$ is defined by the formula $$\forall x \in G: \qquad \Delta(x) = |\! \det(\operatorname{Ad} x)|,$$ where $\operatorname{Ad}$ is the adjoint representation.

Examples.

1. The Haar measure on the additive group $\mathbf{R}$ and on the quotient group $\mathbf{R} / \mathbf{Z}$ (the group of rotations of the circle) is the same as the ordinary Lebesgue measure.
2. The general linear group $\operatorname{GL}(n,\mathbf{F})$, where $\mathbf{F} \in \{ \mathbf{R},\mathbf{C} \}$, is unimodular, and the Haar measure has the form

$$\mathrm{d}{\mu(x)} = |\! \det(x)|^{- k} ~ \mathrm{d}{x},$$ where $k = n$ for $\mathbf{F} = \mathbf{R}$ and $k = 2 n$ for $\mathbf{F} = \mathbf{C}$, and $\mathrm{d}{x}$ is the Lebesgue measure on the Euclidean space of all matrices of order $n$ over the field $\mathbf{F}$.

If $G$ is a locally compact group, $H$ is a closed subgroup of it, $X$ is the homogeneous space $G / H$, $\Delta$ and $\delta$ are the modular functions of $G$ and $H$, respectively, and $\chi$ is a continuous homomorphism of $G$ into $\mathbf{R}^{+}$ whose restriction to $H$ is given by the formula $$\forall h \in H: \qquad \chi(h) = \delta(h) \Delta(h^{-1}),$$ then there exists a positive measure $\nu$ on the $\sigma$-ring $T$ of sets $E \subseteq G / H = X$ that is generated by the family of compact subsets of $X$; it is uniquely determined by the condition: $$\forall f \in {C_{c}}(G): \qquad \int_{G / H} \left[ \int_{H} f(g h) ~ \mathrm{d}{\mu(h)} \right] \mathrm{d}{\nu(g)} = \int_{G} f(g) \chi(g) ~ \mathrm{d}{\mu(g)},$$ where $g = g H \in X$, and $$\forall h \in {C_{c}}(X): \qquad \int_{X} h(g^{-1} x) ~ \mathrm{d}{\nu(x)} = \chi(g) \int_{X} h(x) \mathrm{d}{\nu(x)}.$$

How to Cite This Entry:
Haar measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Haar_measure&oldid=41190
This article was adapted from an original article by D.P. ZhelobenkoA.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article