Haar measure

In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups.

This measure was introduced by Alfréd Haar, a Hungarian mathematician, about 1932. Haar measures are used in many parts of analysis and number theory.

Let G be a locally compact topological group. In this article, the σ-algebra generated by all compact subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If a is an element of G and S is a subset of G, then the we define the left and right translates of S as follows:

• Left translate:

$aS=\{a\cdot s:s\in S\}.$

• Right translate:

$Sa=\{s\cdot a:s\in S\}.$

Left and right translates map Borel sets into Borel sets.

A measure μ on the Borel subsets of G is called left-translation-invariant if and only if for all Borel subsets S of G and all a in G one has

$\mu (aS)=\mu (S).\quad$

A similar definition is made for right translation invariance.

Existence of Haar measure

It turns out that there is, up to a positive multiplicative constant, only one left-translation-invariant countably additive regular measure μ on the Borel subsets of G such that μ(U) > 0 for any open non-empty Borel set U. Here, following Halmos, Section 52, we say μ is regular iff:

• μ(K) is finite for every compact set K.

• Every Borel set E is outer regular:

$\mu (E)=\inf\{\mu (U):E\subseteq U,U{\mbox{ open and Borel}}\}.$

• If E is Borel, then E is inner regular:

$\mu (E)=\sup\{\mu (K):K\subseteq E,K{\mbox{ compact }}\}.$

Remark. Note that in some pathological cases, a set can be open without being Borel. For this reason, in the property of outer regularity, the range of the infimum is specifically stated to be over sets which are open and Borel. These pathologies never occur if G is a locally compact group whose underlying topology is separable metric; note that in this case the Borel structure is that generated by all open sets.

It can also be proved that there exists an essentially unique right-translation-invariant Borel measure, but the two measures need not coincide. Indeed the left and right invariance are equivalent only for so-called unimodular groups (see beleow). Using the general theory of Lebesgue integration, one can then define an integral for all Borel measurable functions f on G. This integral is called the Haar integral. If μ is a left Haar measure, then

$\int _{G}f(sx)\ d\mu (x)=\int _{G}f(x)\ d\mu (x)$

for any integrable function f. This is immediate for step functions being essentially the definition of left invariance.

Note that if μ is a left Haar measure, then

$\nu (S)=\mu (S^{-1})\quad$

is a right Haar measure. To show right invariance, apply the definition:

$\nu (Sa)=\mu ((Sa)^{-1})=\mu (a^{-1}S^{-1})=\mu (S^{-1})=\nu (S).\quad$

This measure is used in harmonic analysis on arbitrary locally compact groups. See Pontryagin duality. A frequently used technique for showing existence of Haar measure on a locally compact group G is showing the existence of a left invariant Radon measure on G.

Note that, unless G is a discrete group, it is impossible to define a countably additive right invariant measure on all subsets of G, assuming the axiom of choice. See non-measurable sets

Examples

• The Haar measure on the topological group (R, +) which takes the value 1 on the interval [0,1] is equal to the restriction of Lebesgue measure to the Borel subsets of R. This can be generalized for (Rⁿ, +).

• If G is the group of positive real numbers with multiplication as operation, then the Haar measure μ(S) is given by

$\mu (S)=\int _{S}{\frac {1}{t}}\,dt$

for any Borel subset S of the positive reals.

This generalizes to the following:

• For G=GL(n,R) left and right Haar measures are proportional and

$\mu (S)=\int _{S}{1 \over |\det(X)|^{n}}\,dX$

where dX denotes the Lebesgue measure on R^{$n^{2}$}, the set of all $n\times n$-matrices. This follows from the change of variables formula.

• More generally, on any Lie group of dimension d a left Haar measure can be associated with any non-zero left-invariant d-form ω, as the Lebesgue measure |ω|; and similarly for right Haar measures. This means also that the modular function can be computed, as the absolute value of the determinant of the adjoint representation.

The modular function

Note that the left translate of a right Haar measure is a right Haar measure. More precisely, if μ is a right Haar measure, then

$A\mapsto \mu (t^{-1}A)\quad$

is a also right invariant. Thus, there exists a unique function Δ such that for every Borel set A

$\mu (t^{-1}A)=\Delta (t)\mu (A).\quad$

A group is unimodular iff the modular function is identically 1. Examples of unimodular groups are compact groups, abelian groups. An example of a non unimodular group is the ax+b group of transformations of the form

$x\mapsto ax+b\quad$

on the real line.

References

• P. Halmos, Measure Theory, D. van Nostrand and Co., 1950.
• Lynn Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand and Co., 1953.
• Andre Weil, Basic Number Theory, Academic Press, 1971