Haar Measure - The Right Haar Measure

The Right Haar Measure

It can also be proved that there exists a unique (up to multiplication by a positive constant) right-translation-invariant Borel measure ν satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure μ. The left and right Haar measures are the same only for so-called unimodular groups (see below). It is quite simple, though, to find a relationship between μ and ν.

Indeed, for a Borel set S, let us denote by the set of inverses of elements of S. If we define

then this is a right Haar measure. To show right invariance, apply the definition:

Because the right measure is unique, it follows that μ_-1 is a multiple of ν and so

for all Borel sets S, where k is some positive constant.