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.

If G is a locally compact topological group, we can consider the σ-algebra X generated by all compact subsets of G. If a is an element of G and S is a set in X, then the set aS = {as : s in S} (where the multiplication is the group operation in G) is also in X. A measure μ on X is called left-translation-invariant if μ(aS) = μ(S) for all a and S.

It turns out that there is, up to a multiplicative constant, only one left-translation-invariant measure on X which is finite on all compact sets. This is the Haar measure on G. (There is also an essentially unique right-translation-invariant measure on X, but the two measures need not coincide.) Using the general Lebesgue integration approach, one can then define an integral for all measurable functions f : G -> R (or C), called the Haar integral. This is the beginning of harmonic analysis.

The Haar measure on the topological group (R, +) which takes the value 1 on the interval [0,1] is equal to the Borel measure. 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 ∫_S 1/x dx for any Borel set S.