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For $G$ a locally compact topological group, a Haar measure on $G$ is a nonnegative, countably additive, real-valued measure on $G$ which is invariant under left translation on $G$. Any such measure is also invariant under right translation on $G$.

A Haar measure always exists and is unique up to multiplication by a positive scalar. If $G$ is compact, then the normalized Haar measure on $G$ is the unique Haar measure on $G$ under which $G$ has total measure 1.

As a special case, if $G$ is finite of order $n$, then the normalized Haar measure is the uniform measure that assigns to each element the measure $1/n$.

Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2022-07-07 09:08:29
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