In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.
Preliminaries Let ( G , ⋅ ) {\displaystyle (G,\cdot )} be a locally compact Hausdorff topological group. The σ {\displaystyle \sigma } -algebra generated by all open subsets of G {\displaystyle G} is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g {\displaystyle g} is an element of G {\displaystyle G} and S {\displaystyle S} is a subset of G {\displaystyle G} , then we define the left and right translates of S {\displaystyle S} by g {\displaystyle g} as follows:
Left translate: g S = { g ⋅ s : s ∈ S } . {\displaystyle gS=\{g\cdot s\,:\,s\in S\}.}
Right translate: S g = { s ⋅ g : s ∈ S } . {\displaystyle Sg=\{s\cdot g\,:\,s\in S\}.}
Left and right translates map Borel sets onto Borel sets. A measure μ {\displaystyle \mu } on the Borel subsets of G {\displaystyle G} is called left-translation-invariant if for all Borel subsets S ⊆ G {\displaystyle S\subseteq G} and all g ∈ G {\displaystyle g\in G} one has
μ ( g S ) = μ ( S ) . {\displaystyle \mu (gS)=\mu (S).}
A measure μ {\displaystyle \mu } on the Borel subsets of G {\displaystyle G} is called right-translation-invariant if for all Borel subsets S ⊆ G {\displaystyle S\subseteq G} and all g ∈ G {\displaystyle g\in G} one has
μ ( S g ) = μ ( S ) . {\displaystyle \mu (Sg)=\mu (S).}
Haar's theorem There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure μ {\displaystyle \mu } on the Borel subsets of G {\displaystyle G} satisfying the following properties:
The measure μ {\displaystyle \mu } is left-translation-invariant: μ ( g S ) = μ ( S ) {\displaystyle \mu (gS)=\mu (S)} for every g ∈ G {\displaystyle g\in G} and all Borel sets S ⊆ G {\displaystyle S\subseteq G} . The measure μ {\displaystyle \mu } is finite on every compact set: μ ( K ) < ∞ {\displaystyle \mu (K)<\infty } for all compact K ⊆ G {\displaystyle K\subseteq G} . The measure μ {\displaystyle \mu } is outer regular on Borel sets S ⊆ G {\displaystyle S\subseteq G} : μ ( S ) = inf { μ ( U ) : S ⊆ U , U open } . {\displaystyle \mu (S)=\inf\{\mu (U):S\subseteq U,U{\text{ open}}\}.}
The measure μ {\displaystyle \mu } is inner regular on open sets U ⊆ G {\displaystyle U\subseteq G} : μ ( U ) = sup { μ ( K ) : K ⊆ U , K compact } . {\displaystyle \mu (U)=\sup\{\mu (K):K\subseteq U,K{\text{ compact}}\}.}
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