Abstract. Let $(S,\Sigma,\lambda )$ be the usual Lebesgue measure space of the unit interval $S, X$ a real Banach space and $\mu $ a vector measure on $\Sigma $ into $X$ absolutely continuous with respect to $\lambda $. If the associated integration map $T$ of $\mu $ extends to and is bounded on $L^p(\lambda )$ for some finite $p$ then every weakly null sequence in the (closed convex hull of the) range of $\mu $ admits a subsequence in $\ell ^2_{\rm weak}(X)$. This is only a sufficient condition; also, in general we cannot do better than $\ell ^2_{\rm weak}(X)$. Further, in every infinite dimensional Banach space $X$ the measures $\mu $ whose $T$'s do not extend in this manner form a residual set in the Banach space $cca(\lambda )$ of all measures from $\Sigma $ into $X$ with relatively norm compact ranges, absolutely continuous with respect to $\lambda $, under the semi variation norm.

AMS Subject Classification (1991): 46G10, 28B45

Received June 26, 2003, and in revised form August 22, 2003. (Registered under 5806/2009.)