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 we define the left and right translates of S as follows:
- Left translate:
- Right translate:
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
A similar definition is made for right translation invariance.
Existence of the left 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:
- If E is Borel, then E is inner regular:
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 right Haar measure
It can also be proved that there exists an essentially unique 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 the set of inverses of elements of S. If we define
then this is a right Haar measure. To show right invariance, apply the definition:
Because the right measure is unique, it follows that μ-1 is a multiple of ν and so
for all Borel sets S, where k is some positive constant.
The Haar integral
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
for any integrable function f. This is immediate for step functions being essentially the definition of left invariance.
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.
- 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
The left translate of a right Haar measure is a right Haar measure. More precisely, if μ is a right Haar measure, then
is also right invariant. Thus, there exists a unique function Δ called the modular function such that for every Borel set A
A group is unimodular iff 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
on the real line.
- Paul Halmos, Measure Theory, D. van Nostrand and Co., 1950.
- Lynn Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand and Co., 1953.
- André Weil, Basic Number Theory, Academic Press, 1971