Spaces of scalar functions that are integrable in the sense of Bartle-Dunford-Schwartz integration, with respect to a~convexly bounded vector measure $\mu$, are studied. For instance, under the assumption that the range space $X$ of $\mu$ is sequentially complete, the effect of the Orlicz-Pettis property (with respect to a weaker topology on $X$) on the size of $L^1(\mu)$ is investigated. Some completeness properties of the space $L^1_\bullet(\mu)$ of `scalarly integrable functions' are established for general $X$.