If is a topological group, a Haar measure is a translation-invariant measure on the Borel sets of . The archetypal example of Haar measure is the Lebesgue measure on the (additive group underlying) cartesian space .
The proper generality in which to discuss Haar measure is where the topological group is assumed to be locally compact Hausdorff, and from here on we assume this. (For topological groups, the Hausdorff assumption is rather mild; it is equivalent to the separation condition. See the discussion at uniform space.)
Let denote the vector space of continuous real-valued functionals with compact support on . This is a locally convex topological vector space where the locally convex structure is specified by the family of seminorms
ranging over compact subsets of . Recall that a Radon measure on may be described as a continuous linear functional
which is positive in the sense that whenever . This defines a measure on the -algebra of Borel sets in the usual sense of measure theory, where
By abuse of notation, we generally conflate and .
A left Haar measure on is a nonzero Radon measure such that
for all and all Borel sets .
Any locally compact Hausdorff topological group admits a Haar mesaure that is unique up to scalar multiple. This result was first proven by Weil. A proof by be found in these online notes by Rubinstein-Salzedo.
The left and the right Haar measure may or may not coincide, groups for which they coincide are called unimodular. Consider the matrix subgroup
The left and right invariant measures are, respectively,
and so G is not unimodular.