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

Contents

Preliminaries

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.

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