Let \(E\) be a separable Banach space, \(X\) a Borel subset of \(E\), and \(\mu\) a non atomic \(\sigma\)-finite Borel measure on \(X\). The author extends to this context a theorem of \textit{C. R. Hobby} and \textit{J. R. Rice} [Proc. Am. Math. Soc. 16, 665-670 (1965; Zbl 0142.09802)]. If \(G\) is an \(m\) - dimensional subspace of \(L_{1}(X,\sum,\mu)\) then there exist \(l\in E^{\ast}\) and the numbers \(-\infty =x_{0}