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 ([1]) (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 $$ \forall h \in {C_{c}}(X): \qquad \int_{X} h(g^{-1} x) ~ \mathrm{d}{\nu(x)} = \chi(g) \int_{X} h(x) \mathrm{d}{\nu(x)}. $$

References

[1]	A. Haar, “Der Massbegriff in der Theorie der kontinuierlichen Gruppen”, Ann. of Math. (2), 34 (1933), pp. 147–169.
[2]	N. Bourbaki, “Elements of mathematics. Integration”, Addison-Wesley (1975), Chapt. 6–8. (Translated from French)
[3]	A. Weil, “L’intégration dans les groupes topologiques et ses applications”, Hermann (1940).
[4]	L.H. Loomis, “An introduction to abstract harmonic analysis”, v. Nostrand (1953).
[5]	S. Helgason, “Differential geometry and symmetric spaces”, Acad. Press (1962).

References

[a1]	E. Hewitt, K.A. Ross, “Abstract harmonic analysis”, 1–2, Springer (1979).