Encyclopedia > Haar measure

  Article Content

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). 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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Sanskrit language

... region circa 1500 BCE. There seems to be no clear evidence that these hordes destroyed the existing Indus Valley culture. (See Aryan invaders). But it is clear tha ...

This page was created in 25 ms