Continuously translating vector-valued measures (original) (raw)

1980, Transactions of the American Mathematical Society

Let G be a locally compact group and A an arbitrary Banach space. L p ( G , A ) {L^p}(G,A) will denote the space of p-integrable A-valued functions on G. M ( G , A ) 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 ) {L^p}(G,A) . Using this result, we prove that if μ ∈ M ( G , A ) \mu \, \in \, M(G,A) , such that either x → μ x x\, \to \, {\mu _x} or x → x μ x{ \to _x}\mu is continuous, then μ ∈ L 1 ( G , A ) \mu \, \in \, {L^1}(G,A) .