Exercise 16, § 3

The notations and conventions are those of Exercise 15.

a) Now prove that, in order that a bounded measure $\mu$ on $G$ be such that $\|\mu * f\|_1 = \|f\|_1$ for every $f \in L^1(G, \beta)$, it is necessary and sufficient that $\mu$ be a point measure of norm 1. (Using the fact that the endomorphism $\gamma(|\mu|)$ of $L^1(G, \beta)$ has norm $\|\mu\|$ (Exercise 15), now prove that it is necessary to have, for every function $f \in \mathscr{K}(G)$,

deduce first that $\tilde{\mu} = c|\mu|$, where $c$ is a constant of absolute value 1, then that $\mu$ is pointwise).

b) Take for $G$ the cyclic group of order 3. Give an example of a non-point measure $\mu$ on $G$ such that $\|\mu * f\|_2 = \|f\|_2$ for every numerical function $f$ defined on $G$.