Exercise 13, § 3

A C-set in $\widehat{G}$ is called a closed subset E of $\widehat{G}$ possessing the following property: if $f\in L^1(G)$, if $\mathscr{F}(f)$ vanishes on E, and if $\varepsilon >0$, there exists a function $g\in L^1(G)$ such that $\|f-f*g\|\leqslant \varepsilon$ and such that Supp$(\mathscr{F}(g))$ is compact and disjoint from E.

a) If E is a C-set in $\widehat{G}$, there exists only one closed ideal I of $L^1(G)$ such that V(I) = E.

b) Every set $\widehat{G}$ reduced to one element is a C-set.

c) The union of two C-sets is a C-set.

d) Every set obtained by translation from a C-set is a C-set.

e) Let H be a closed subgroup of $\widehat{G}$, E a closed subset of H, $E'$ the boundary of E relative to H. If $E'$ is a C-set relative to $\widehat{G}$, then E is a C-set relative to $\widehat{G}$. (Use Exer. 21 of II, p. 270.) f) In $\mathbf{R}^n$, every intersection E of a finite number of closed half-spaces is a C-set. (Reason by induction on the dimension of the affine subspace generated by E, using b), c), d), e).)