Haar measure

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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.

If 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 right and left translates of S as follows:

Left translate:

Right translate:

Right and left 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,

A similar definition is made for right translation invariance.

Table of contents

1 Existence of Haar measure
2 Examples
3 The modular function
4 References

Existence of Haar measure

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

μ(K) is finite for every compact set K.
For every Borel set E, is outer regular:

If E is Borel, the E is inner regular:

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.

Theorem. If G is a locally compact group, then there is a unique regular left-translation-invariant measure Borel measure on G.

This is the left Haar measure on G. There is also an essentially unique right-translation-invariant Borel measure, but the two measures need not coincide. Indeed they are related by the modular function on the group. Using the general theory of Lebesgue integration approach, 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

for iny integrable function f. This is immediate for step functions being essentially the definition of left invariance. From this follows that

is a right Haar integral.

This integral 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 it is impossible to define a countably additive right invariant measure on all subsets of G for all but discrete subgroups, assuming that is 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

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

where dX denotes the Lebesgue measure on R, the set of all -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

We need to show how right and left Haar measures are related. Note that the left translate of a right Haar measure (or integral) is a Haar measure (or integral): More precisely, if μ is a right Haar measure,

is a also right invariant. Thus there is unique function such that

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