Haar measure on a Lie group

Summary

On a Lie group, the Haar measure can be written down explicitly as the measure defined by a nonzero left-invariant differential form of top degree. This gives a smooth density in every chart, so integrals over the group can be computed with ordinary calculus. The left and right Haar measures agree exactly when $\abs{\det \operatorname{Ad}(g)} = 1$ for all $g$, which holds for compact, abelian and semisimple groups. It is the basic tool for averaging over a group, as in the proof that representations of compact groups are unitarisable.