I am taking a seminar that follows Serre's book "Linear Representations of Finite Groups", and I am preparing a talk on Chapter 7 on induced representations (Frobenius reciprocity, Mackey's formula for $Res_H Ind^G_H(W)$ and Mackey's irreducibility criterion).
Serre says on p. 34 that one can induce a representation from a finite index closed subgroup $H$ of a compact group $G$ to $G$ in the same way as for $G$ finite. Why is this true? I am wondering whether also the results from Chapter 7 carry over to $G$ compact?
Is there a reference for the representation theory of compact groups in general?
Thank you!