Uniform $\sigma$-additivity in spaces of Bochner or Pettis integrable functions over a locally compact group

Abstract:

If $G$ is an abelian locally compact group with Haar measure $\mu$, $E$ is a Banach space and $K \subset L_E^1(\mu )$, we give necessary and sufficient conditions for the set $\left \{ {{f_{( \cdot )}}\left | f \right |d\mu ;f \in K} \right \}$ to be uniformly $\sigma$-additive in terms of uniform convergence on $K$, for the topology $\sigma (L_E^1,L_{E’}^\infty )$ of convolution and translation operators. In case $E = R$, this gives a new characterization of relatively weakly compact sets $K \subset {L^1}$.