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. 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. 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 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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