Exercise 21, § 1

Let H be a closed subgroup of G and $f$ an element of $L^1(G)$ such that $\mathscr{F}(f)$ vanishes on $H^{\bot}$. Let $\varepsilon >0$. There exists a measure $\mu\in \mathscr{M}^1(G)$ concentrated on H such that $\|\mu\|\leqslant 2,\|f*\mu\|\leqslant \varepsilon$ and such that $\mathscr{F}(\mu) = 1$ in a neighbourhood of $H^{\bot}$.