Continuously translating vector-valued measures

Let G be a locally compact group and A an arbitrary Banach space. ${L^p}(G,A)$ will denote the space of p-integrable A-valued functions on G. $M(G,A)$ will denote the space of regular A-valued Borel measures of bounded variation on G. In this paper, we characterise the relatively compact subsets of ${L^p}(G,A)$. Using this result, we prove that if $\mu \in M(G,A)$, such that either $x \to {\mu _x}$ or $x{ \to _x}\mu$ is continuous, then $\mu \in {L^1}(G,A)$.