nLab
Haar measure

Haar measure

Idea

If GG is a topological group, a Haar measure is a translation-invariant measure on the Borel sets of GG. The archetypal example of Haar measure is the Lebesgue measure on the (additive group underlying) cartesian space n\mathbb{R}^n.

Definition

The proper generality in which to discuss Haar measure is where the topological group GG is assumed to be locally compact Hausdorff, and from here on we assume this. (For topological groups, the Hausdorff assumption is rather mild; it is equivalent to the T 0T_0 separation condition. See the discussion at uniform space.)

Let C c(G)C_c(G) denote the vector space of continuous real-valued functionals with compact support on GG. This is a locally convex topological vector space where the locally convex structure is specified by the family of seminorms

ρ K(f)=sup xK|f(x)|,\rho_K(f) = \sup_{x \in K} |f(x)|,

KK ranging over compact subsets of GG. Recall that a Radon measure on GG may be described as a continuous linear functional

μ:C c(G)\mu: C_c(G) \to \mathbb{R}

which is positive in the sense that μ(f)0\mu(f) \geq 0 whenever f0f \geq 0. This defines a measure μ^\hat{\mu} on the σ\sigma-algebra of Borel sets in the usual sense of measure theory, where

μ^(B)=sup{μ(f):supp(f)=KB,ρ K(f)=1}\hat{\mu}(B) = sup \{\mu(f): supp(f) = K \subseteq B, \rho_K(f) = 1\}

By abuse of notation, we generally conflate μ\mu and μ^\hat{\mu}.

A left Haar measure on GG is a nonzero Radon measure μ\mu such that

μ(gB)=μ(B)\mu(g B) = \mu(B)

for all gGg \in G and all Borel sets BB.

Existence and Uniqueness

Any locally compact Hausdorff topological group GG admits a Haar mesaure that is unique up to scalar multiple. This result was first proven by Weil. A proof by be found in these online notes by Rubinstein-Salzedo.

Left and Right Haar Measures that Differ

The left and the right Haar measure may or may not coincide, groups for which they coincide are called unimodular. Consider the matrix subgroup

G:={(y x 0 1)|x,y,y>0} G := \left\{ \left.\, \begin{pmatrix} y & x \\ 0 & 1 \end{pmatrix}\,\right|\, x, y \in \mathbb{R}, y \gt 0 \right\}

The left and right invariant measures are, respectively,

μ L=y 2dxdy,μ R=y 1dxdy \mu_L = y^{-2} \,\mathrm{d}x \,\mathrm{d}y,\quad \mu_R = y^{-1} \,\mathrm{d}x \,\mathrm{d}y

and so G is not unimodular.

Abelian groups are obviously unimodular; so are compact groups and discrete groups.

Revised on June 15, 2013 09:32:37 by Toby Bartels (98.23.135.113)