Haar measure and integration

Choose a nonzero top-degree density at the identity of a finite-dimensional Lie group and transport it by left translations. Integrating that smooth positive density gives a nonzero left-invariant Radon measure, finite on compact sets. Haar's uniqueness theorem says that any two left Haar measures differ by a positive constant factor. The more general existence theorem applies to locally compact Hausdorff groups, but the invariant-density construction is the concrete Lie group version used here.

A compact group has finite positive Haar volume and can therefore be normalized to total mass one. That normalized measure is also right invariant. One way to see this is that a right translate of a left Haar measure is another left Haar measure, and its total mass has not changed. Noncompact groups generally cannot use this probability normalization: a nonzero Haar measure has finite total mass exactly when the group is compact. Uniform group averages and finite probability averages are therefore not interchangeable in noncompact settings.

The positive function Delta is a continuous group homomorphism into the multiplicative positive reals. A group is unimodular when it is identically one, meaning that left Haar measure is also right invariant. For a Lie group with the displayed convention, Delta-g is the inverse absolute determinant of Ad-g. Along exp tX this becomes the exponential of minus t times the trace of ad-X. This formula links infinitesimal volume distortion to finite conjugation and gives a convenient independent check on coordinate calculations.

Abelian groups are unimodular because their adjoint action is trivial. Compact groups are unimodular because the image of a compact group under Delta is a compact subgroup of the positive reals, hence trivial. Connected semisimple Lie groups are also unimodular because their infinitesimal adjoint maps have zero trace. None of these facts says that an arbitrary noncompact matrix group has its ordinary Lebesgue measure as Haar measure. A coordinate-dependent density is often essential.

For the positive affine group, ordinary parameter area changes under both left and right multiplication. A density proportional to one over the square of the scale corrects left multiplication, while one over the scale corrects right multiplication. The two differ, and the group's modular function records that difference. Translating a coordinate rectangle while keeping its unweighted area fixed is therefore not an invariant-volume test. Both the coordinate Jacobian and the density at the transformed coordinates must be included.

On a compact group, normalized Haar measure allows averaging an inner product or a function without choosing a preferred group element. For a finite-dimensional continuous representation, averaging a positive inner product gives an invariant positive inner product. This is one reason compact-group representation theory is especially well behaved. On a noncompact group that same unweighted integral may diverge, so an argument requiring a normalized invariant average must retain the compactness hypothesis or provide a different construction.