Exercise 7, § 3

Let G be a locally compact group.

a) Show that if a positive measure $\mu \in \mathscr{C}'(G)$ admits an inverse in $\mathscr{C}'(G)$ that is a positive measure, then $\mu$ is necessarily a point measure.

b) Taking G to be the finite group $\mathbf{Z}/2\mathbf{Z}$, give an example of a non point measure $\mu \in \mathscr{M}(G)$ that is invertible in $\mathscr{M}(G)$.