Exercise 16, § 4

Notations and conventions are those of Exer. 15.

a) Show that, for a bounded measure $\mu$ on G to be such that $\| \mu * f \|_1 = \| f \|_1$ for all $f \in L^1(G, \beta)$, 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 $\| \mu \|$ (Exer. 15), show that one necessarily has, for every function $f \in \mathscr{K}(G)$,

from this, first deduce that $\mu = c |\mu|$, where c is a constant of absolute value 1, then that $\mu$ is a point measure.)

b) Take for G the cyclic group of order 3. Give an example of a measure $\mu$ on G, not a point measure, such that $\| \mu * f \|_2 = \| f \|_2$ for every numerical function $f$ defined on G.