Haar measure

Haar measure
Name	Haar measure
Type	Invariant measure
Field	Measure theory; Lebesgue measure; Harmonic analysis
Introduced	1933
Introduced by	Alfréd Haar
Primary sources	Haar's original paper

Haar measure Haar measure is a translation-invariant measure on locally compact topological groups that enables integration compatible with the group structure. It underpins constructions in Harmonic analysis, Representation theory, and probability on groups, connecting classical measures such as Lebesgue measure on R^n and counting measure on discrete groups. The concept, introduced by Alfréd Haar, provides a canonical way to integrate continuous or compactly supported functions equivariantly with respect to group actions.

Definition and basic properties

A Haar measure on a locally compact topological group G is a nonzero regular Borel measure μ on G that is left-invariant: for every measurable set A and every g in G, μ(gA)=μ(A). Fundamental properties include uniqueness up to positive scalar multiples, regularity in the sense used in Measure theory (inner regularity on open sets and outer regularity on all Borel sets), and σ-finiteness on σ-compact groups such as R^n, the torus, and SO(3). For compact groups like SU(2) and O(n), Haar measure can be normalized to a probability measure; for discrete groups such as Z and free groups, Haar measure coincides with counting measure. Left and right Haar measures may differ in non-abelian settings, giving rise to the modular function associated with groups like ax+b group or Borel subgroup of SL(2,R).

Existence and uniqueness theorem

The Haar existence and uniqueness theorem asserts that every locally compact group G admits a nonzero left-invariant regular Borel measure, and any two such measures are proportional. The proof strategy historically used compactness arguments and the Riesz representation theorem as in work following Alfréd Haar; modern expositions employ the Banach–Alaoglu theorem and fixed-point arguments reminiscent of techniques used in John von Neumann's functional analytic investigations and in the development of Amenability theory by John von Neumann and Murray and von Neumann. Uniqueness up to scaling is typically shown using the regularity and inner approximation by compact sets, paralleling arguments used in classical existence proofs for Lebesgue measure on R^n.

Construction methods

Constructions of Haar measure vary by context. For compact groups such as U(n), one can average any regular measure by convolution with point masses using the Peter–Weyl theorem techniques developed by Hermann Weyl; for second-countable groups continuity arguments tied to C*-algebra representations and the Gelfand–Naimark framework yield Haar measure from invariant states. For locally compact abelian groups, Pontryagin duality due to Lev Pontryagin facilitates explicit constructions via Fourier transform methods employed in Abstract harmonic analysis by Lloyd Schwartz and Marshall Stone. For discrete groups Haar measure is counting measure, while for Lie groups like GL(n,R) and SL(2,R) one constructs left-invariant volume forms from Haar measures derived from left-invariant top-degree differential forms and the theory of Differential geometry on manifolds initiated by Élie Cartan.

Examples and computations

Classic examples include R^n with Lebesgue measure, compact groups like SO(3) with normalized invariant volume, and finite groups where Haar measure is uniform distribution over group elements relevant to Cayley graph random walks. For matrix groups, Haar measure on GL(n,R) can be written using Lebesgue measure on matrix entries multiplied by |det A|^{-n} factors after appropriate coordinate changes; on SL(2,R) explicit formulas use coordinates from the Iwasawa decomposition linked to Harish-Chandra's work. Computations for semidirect products like the ax+b group reveal nontrivial modular functions, and for p-adic groups such as GL(n,Q_p), Haar measure arises from the unique translation-invariant measure on the totally disconnected locally compact topology central to Adeles and Tate's thesis.

Applications in harmonic analysis and representation theory

Haar measure is central to the formulation of the Fourier transform on locally compact abelian groups in Pontryagin duality and to the Plancherel theorem for groups such as R^n and the circle group. In non-abelian harmonic analysis it underlies the definition of unitary representations of groups like SU(2) and SL(2,R), enabling the decomposition of regular representations via the Peter–Weyl theorem and the Plancherel formula developed by Harish-Chandra for reductive Lie groups. Haar measure also plays a critical role in ergodic theory on spaces with actions by groups such as SL(2,Z), in the study of random walks on groups informed by work of Hillel Furstenberg, and in the construction of group C*-algebras in the framework of Gelfand–Naimark theory.

Invariant integration and modular function

Invariant integration with respect to Haar measure provides a left-invariant integral ∫_G f(g) dμ(g) used throughout harmonic analysis and probability on groups. When left- and right-invariance differ, the modular function Δ: G → (0,∞) measures the ratio between right- and left-Haar measures and is a continuous group homomorphism; for unimodular groups such as SO(n), SU(n), and SL(2,C), Δ≡1. The modular function enters the formulation of the group von Neumann algebra and the theory of induced representations developed by George Mackey, and appears explicitly in integration formulas like the Weyl integration formula used in the representation theory of compact Lie groups pioneered by Hermann Weyl.

Category:Measure theory