Plancherel measure

Summary

The Plancherel measure of a second countable unimodular locally compact group of type I is a measure on its unitary dual. It tells how the regular representation on $L^2(G)$ is built from irreducible representations. With it, the $L^2$ norm of a function equals an integral of the Hilbert–Schmidt norms of its Fourier transform. Its support is the reduced dual, the set of irreducible representations weakly contained in the regular representation on $L^2(G)$.