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weak* measurable function; copy of $c_0$; copy of $\ell_1$
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 $c_{0}$.
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