Exercise 3, § 2

Let $G =\mathbf{R}$ and let H be the group $\mathbf{R}$ endowed with the discrete topology and the counting measure, which is a Haar measure.

a) For every $t\in G$, and every $f\in L^2$(H), the mapping $\varrho (t)f:x\mapsto f(x+t)$ belongs to $L^2(H)$; the mapping $\varrho (t)$ is a unitary mapping on $L^2(H)$. b) For all $f$ and $g$ in $L^2$(H), the mapping from G into $\mathbf{C}$ defined by $t\mapsto \langle f|\varrho (t)g\rangle$ is measurable. c) Let $f$ and $g$ be in $L^2(H)$. If there exists $t\in \mathbf{R}$ such that $\langle f|\varrho (t)g\rangle$ is nonzero, then the mapping $t\mapsto \langle f|\varrho (t)g\rangle$ from G into $\mathbf{C}$ is not continuous. (In particular, the hypothesis that E is of countable type cannot be omitted from Cor. 2 of INT, VIII, p. 171, §4, n$^o7.$)

d) There exists $f\in L^2(H)$ such that the subspace of $L^2(H)$ generated by the elements $\varrho (t)f$, where $t\in \mathbf{R}$, is dense in $L^2(H)$.