I would like to understand what the Plancherel measure is on the dual of a compact group, in more explicit terms than with an implicit definition. Indeed, to the extent of my knowledge it is defined as the only positive Radon measure such that a certain inversion formula holds, namely $$\int_{\hat{G}} f(\pi)d\mu(\pi) = f(1).$$

What can be said more precisely when a representation is given explicitly? For instance, in the case of finite dimension supercuspidal representations, do we have that $$\mu(\pi) = \dim(\pi)^{-1},$$

as in the finite case, or something similar? Are there references constructing from scratch the Plancherel measure and manipulating it?