Abstract:If $(\Omega ,\Sigma ,\mu )$ is a finite measure space and $X$ a Banach space, in this note we show that $L_{w^{\ast }}^{1}(\mu ,X^{\ast })$, the Banach space of all classes of weak* equivalent $X^{\ast }$-valued weak* measurable functions $f$ defined on $\Omega $ such that $\|f(\omega )\| \leq g(\omega )$ a.e. for some $g\in L_{1}(\mu )$ equipped with its usual norm, contains a copy of $c_{0}$ if and only if $X^{\ast }$ contains a copy of $c_{0}$.
Keywords: weak* measurable function, copy of $c_{0}$, copy of $\ell _{1}$
AMS Subject Classification: 46G10, 46E40