On control measures of multimeasures

Let $M$ be a multimeasure defined on a $\sigma$-algebra and taking values in the family of bounded non-empty subsets of a Banach space $X$. We prove that $M$ admits a control measure whenever $X$ contains no subspace isomorphic to $c_0({\omega }_1)$. The additional assumption on $X$ is shown to be essential.