A left Haar measure on a locally compact Hausdorff topological group is a nonzero left invariant Borel measure which is finite on compact sets, outer regular on Borel sets, and inner regular on open sets. Every such group has a left Haar measure, unique up to a positive multiplicative constant. If the group is Abelian or compact, then this measure is also right invariant.
More formally, let 
be a locally compact Hausdorff topological
group. Then a left invariant Haar measure on 
is a Borel measure 
satisfying the following conditions:
1. 
for every 
and every Borel set 
.
2. 
for every nonempty open
set 
.
3. 
for every compact
set 
.
4. For every Borel set 
, 
is the infimum of 
over open sets 
containing 
.
5. For every open set 
, 
is the supremum of 
over compact sets 
contained in 
.
For example, the Lebesgue measure is an invariant Haar measure on real numbers.
In addition, if 
is an (algebraic) group, then 
with the discrete topology
is a locally compact group.
A left invariant Haar measure on 
is the counting measure
on 
.
