Translation continuity in Lp on a locally compact group ($\Vert T_{x}f-f{\Vert }_p⟶0(x⟶e)$)

For left Haar measure, $T_{x}f(t)=f(x^{-1}t)$ is an isometry on $L^p(G)$, $1\le p<\infty$. For compactly supported continuous functions, uniform continuity on a slightly enlarged compact support proves norm continuity. Approximation by those functions and the triangle inequality extend it to all $L^p$. The continuous image of a compact set of translation parameters is consequently compact in $L^p$.