Exercise 14, § 1

a) The space $L^1(G)$ is a closed ideal in $\mathscr{M}^1(G)$, hence $\widehat{G}$ is identified with an open subset of $\mathsf{X}(\mathscr{M}^1(G))$ (cf. Exercise 17 of I, p. 169).

b) Let $\mathscr{M}_d(G)$ (resp. $\mathscr{M}_a(G)$) be the set of $\mu\in \mathscr{M}^1(G)$ which are diffuse (resp. atomic). Then $\mathscr{M}_d(G)$ is a closed ideal of $\mathscr{M}^1(G)$ and $\mathscr{M}_a(G)$ is a closed subalgebra of $\mathscr{M}^1(G)$.