Normalized Haar measure on a compact group

Summary

On a compact group, every Haar measure has finite total mass. The normalized Haar measure is the unique Haar measure of total mass $1$, so it is a probability measure. It is both left- and right-invariant, and it gives the natural notion of the average of a function over the group. It is the measure used in the Peter–Weyl theorem and in the orthogonality relations for characters.