Editing Haar measure (section)

==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 <math> \nu</math> satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the  left-translation-invariant measure <math>\mu</math>. 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 <math>\mu</math> and <math>\nu</math>.

Indeed, for a Borel set <math>S</math>, let us denote by <math>S^{-1}</math> the set of inverses of elements of <math>S</math>. If we define 
:<math> \mu_{-1}(S) = \mu(S^{-1}) \quad </math>
then this is a right Haar measure. To show right invariance,  apply the definition:

:<math> \mu_{-1}(S g) = \mu((S g)^{-1}) = \mu(g^{-1} S^{-1}) = \mu(S^{-1}) = \mu_{-1}(S). \quad </math>

Because the right measure is unique, it follows that <math>\mu_{-1}</math> is a multiple of <math>\nu</math> and so
:<math>\mu(S^{-1})=k\nu(S)\,</math>
for all Borel sets <math>S</math>, where <math>k</math> is some positive constant.

===The modular function===

The ''left'' translate of a right Haar measure is a right Haar measure. More precisely, if <math>\nu</math> is a right Haar measure, then for any fixed choice of a group element ''g'',

:<math> S \mapsto \nu (g^{-1} S) \quad </math>

is also right invariant.  Thus, by uniqueness up to a constant scaling factor of the Haar measure, there exists a function <math>\Delta</math> from the group to the positive reals, called the '''Haar modulus''', '''modular function''' or '''modular character''', such that for every Borel set <math>S</math>

:<math> \nu (g^{-1} S) = \Delta(g) \nu(S). \quad</math>

Since right Haar measure is well-defined up to a positive scaling factor, this equation shows the modular function is independent of the choice of right Haar measure in the above equation.

The modular function is a continuous group homomorphism from ''G'' to the multiplicative group of [[positive real numbers]]. A group is called '''unimodular''' if the modular function is identically <math>1</math>, or, equivalently, if the Haar measure is both left and right invariant. Examples of unimodular groups are [[abelian group]]s, [[compact group]]s, [[discrete group]]s (e.g., [[finite group]]s), [[semisimple Lie group]]s and [[connected space|connected]] [[nilpotent Lie group]]s.{{citation needed|date=June 2018}} An example of a non-unimodular group is the group of affine transformations

:<math>\big\{ x \mapsto a x + b : a\in\R\setminus\{0\}, b\in\R \big\}=\left\{\begin{bmatrix}
 a & b \\
0 & 1 \end{bmatrix}\right\}</math>

on the real line. This example shows that a [[solvable group|solvable]] Lie group need not be unimodular.
In this group a left Haar measure is given by <math>\frac{1}{a^2}da\wedge db</math>, and a right Haar measure by <math>\frac{1}{|a|}da\wedge db</math>.