Plancherel measure on set of irreducible representations of a finite group

Definition

Suppose $G$ is a finite group. Let $\operatorname {Irr} (G)$ be the set of irreducible representations (up to equivalence) of $G$ over the field $\mathbb {C}$ of complex numbers. The Plancherel measure on this set assigns to each element of $\operatorname {Irr} (G)$ the measure $d^{2}/|G|$ where $d$ is the degree of the representation.

The Plancherel measure is a probability measure in the sense that the total measure of the set is $1$. This follows from the fact that sum of squares of degrees of irreducible representations equals group order.