Let $E$ be some Banach space. For $a\in\mathbb{R}^n$ let $\tau_a\colon E^{(\mathbb{R}^n)}\to E^{(\mathbb{R}^n)}$ be the translation given by $$(\tau_a\varphi)(x):=\varphi(x-a),\quad x\in\mathbb{R}^n,\, \varphi\in E^{(\mathbb{R}^n)}.$$
I want to show that, for $p\in[1,\infty[$, we have $\lim_{a\to 0}\Vert \tau_af-f\Vert_p=0$ for all $f\in L_p(\mathbb{R}^n,\lambda_n, E)$, and that if $\lim_{a\to 0}\Vert \tau_af-f\Vert_\infty =0$ we have some bounded, unformly continuous map $g\colon\mathbb{R}^n\to E$ such that $f=g$ $\lambda_n$-a.e. I don't know how.
(I know that, for $p\in [1,\infty]$, $\tau_a$ is a topological automorphism of $L_p(\mathbb{R}^n,\lambda_n, E)$ with $\Vert\tau_a\Vert= 1$. Does this help?)