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.

Preliminaries

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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 We follow the conventions of Halmos' textbook. Many authors instead use the term Borel algebra to denote the σ-algebra generated by the open sets.. 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 we define the left and right translates of S as follows:

• Left translate:

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

• Right translate:

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

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

$\backslash mu(a\; S)\; =\; \backslash mu(S).\; \backslash quad$

A similar definition is made for right translation invariance.

Existence and uniqueness of the left Haar measure

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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. Such a measure is called a left Haar measure. Following Halmos Paul Halmos, Measure Theory, D. van Nostrand and Co., 1950. Section 52, we say μ is regular if and only if:

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

• Every Borel set E is outer regular:

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

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

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

Remark. 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; in this case the Borel structure is that generated by all open sets.

The existence of Haar measure was first proven in full generality by WeilAndré Weil, L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, Hermann, 1940. The special case of invariant measure for compact groups had been shown by Haar in 1933 A. Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Ann. Math., v34 (1933)..

The right Haar measure

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It can also be proved that there exists a unique (up to multiplication by a positive constant) right-translation-invariant Borel measure ν, but it need not coincide with the left-translation-invariant measure μ. These measures are the same only for so-called unimodular groups (see below). It is quite simple though to find a relationship between μ and ν.

Indeed, for a Borel set S, let us denote by $S^\{-1\}$ the set of inverses of elements of S. If we define

$\backslash mu\_\{-1\}(S)\; =\; \backslash mu(S^\{-1\})\; \backslash quad$

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

$\backslash mu\_\{-1\}(S\; a)\; =\; \backslash mu((S\; a)^\{-1\})\; =\; \backslash mu(a^\{-1\}\; S^\{-1\})\; =\; \backslash mu(S^\{-1\})\; =\; \backslash mu\_\{-1\}(S).\; \backslash quad$

Because the right measure is unique, it follows that μ_-1 is a multiple of ν and so

$\backslash mu(S^\{-1\})=k\backslash nu(S)\backslash ,$

for all Borel sets S, where k is some positive constant.

The Haar integral

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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

$\backslash int\_G\; f(s\; x)\; \backslash \; d\backslash mu(x)\; =\; \backslash int\_G\; f(x)\; \backslash \; d\backslash mu(x)$

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

Uses

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The Haar measures are used in harmonic analysis on arbitrary locally compact groups, see Pontryagin duality. A frequently used technique for proving the existence of a Haar measure on a locally compact group G is showing the existence of a left invariant Radon measure on G.

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

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• The Haar measure on the topological group (R, +) which takes the value 1 on the interval ^* 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

$\backslash mu(S)\; =\; \backslash int\_S\; \backslash frac\{1\}\{t\}\; \backslash ,\; dt$

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

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

where dX denotes the Lebesgue measure on R^{$n^2$}, the set of all $n\backslash times\; n$-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

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The left translate of a right Haar measure is a right Haar measure. More precisely, if μ is a right Haar measure, then

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

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

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

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

$x\; \backslash mapsto\; a\; x\; +\; b\backslash quad$

on the real line.

Notes and references

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Additional references

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

See also

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• Alfred Haar
• Haar wavelet

Lie groups | Topological groups | Mathematical analysis | Measure theory

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