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 $1/n$.
- Review status: reviewed
- Last edited by John Jones on 2022-07-07 09:08:29
- 2022-07-07 09:08:29 by John Jones (Reviewed)
- 2019-04-20 14:17:31 by Kiran S. Kedlaya
- 2019-04-20 14:10:28 by Kiran S. Kedlaya