Haar measure

related topics
{math, number, function}
{rate, high, increase}
{group, member, jewish}

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, in 1933.[1] Haar measures are used in many parts of analysis and number theory, and also in estimation theory.



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.[2] 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.

The Haar theorem

There is, up to a positive multiplicative constant, a unique countably additive measure μ on the Borel subsets of G satisfying the following properties:

  • μ(gE) = μ(E) for any g in G and Borel set E (left-translation-invariance).
  • μ(K) is finite for every compact set K.
  • Every Borel set E is outer regular:
  • Every open set E is inner regular:

Such a measure on G is called a left Haar measure. It can be shown as a consequence of the above properties that μ(U) > 0 for every open non-empty set U. In particular, if G is compact then μ(G) is finite and positive, so we can uniquely specify a left Haar measure on G by adding the normalization condition μ(G) = 1.

Full article ▸

related documents
Splitting lemma
Unicity distance
Extended real number line
Richard's paradox
Axiom of pairing
Legendre symbol
Functional analysis
Assignment problem
Elementary group theory
Ring (mathematics)
Meromorphic function
Examples of groups
Presburger arithmetic
Mathematical model
Queue (data structure)
Lagrange inversion theorem
Chain rule
Statistical independence
Extended Backus–Naur Form
Referential transparency (computer science)
Oracle machine
XSL Transformations
Quotient group
Boolean ring