Haar measure
1 Definition of Haar measures
Let $G$ be a locally compact topological group, and denote by $\mathcal{B}$ the sigma algebra generated by the closed compact subsets of $G$. A left Haar measure on $G$ is a measure^{} $\mu $ on $\mathcal{B}$ which is:

1.
outer regular on all sets $B\in \mathcal{B}$

2.
inner regular on all open sets $U\in \mathcal{B}$

3.
finite on all compact sets $K\in \mathcal{B}$

4.
invariant under left translation^{}: $\mu (gB)=\mu (B)$ for all sets $B\in \mathcal{B}$

5.
nontrivial: $\mu (B)>0$ for all nonempty open sets $B\in \mathcal{B}$.
A right Haar measure on $G$ is defined similarly, except with left translation invariance replaced by right translation invariance ($\mu (Bg)=\mu (B)$ for all sets $B\in \mathcal{B}$). A biinvariant Haar measure is a Haar measure that is both left invariant and right invariant.
2 Existence of Haar measures
For any discrete topological group^{} $G$, the counting measure on $G$ is a biinvariant Haar measure. More generally, every locally compact topological group $G$ has a left Haar measure $\mu $, which is unique up to scalar multiples. In addition, $G$ also admits a right Haar measure, and for an abelian group^{} $G$ the left and right Haar measures are always equal. The Haar measure plays an important role in the development of Fourier analysis and representation theory on locally compact groups such as Lie groups^{} and profinite groups.
Title  Haar measure 

Canonical name  HaarMeasure 
Date of creation  20130322 12:40:55 
Last modified on  20130322 12:40:55 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  9 
Author  djao (24) 
Entry type  Definition 
Classification  msc 28C10 
Defines  left Haar measure 
Defines  right Haar measure 
Defines  biinvariant Haar measure 