Consider on \((\mathbb{R}, \mathscr{B}(\mathbb{R}), d x)\) the Lebesgue space \(\mathcal{L}^{p}(d x), \quad 1 \leqslant p<\infty\). We set \(\tau_{h} f(x):=f(x-h), h \in \mathbb{R}\). Show that

(i) \(\tau_{h}\) is an isometry on \(\mathcal{L}^{p}(d x)\),

(ii) \(\lim _{h ightarrow 0}\left\|\tau_{h} f-fight\|_{p}=0\) and \(\lim _{h ightarrow \infty}\left\|\tau_{h} f-fight\|_{p}=2^{1 / p}\|f\|_{p}\).