Control Measures for Bochner $L_{0}$-Valued Vector Measures

Abstract

It is shown that for any finite positive measure $\mu$ defined on a measure space $(S, \Sigma)$, and any Banach or Fr\'echet space $Z$, the control measure Theorem of Talagrand (T) is true for the case when the (stochastic) vector measure $\boldsymbol{m}:\mathcal{E} \to L_0(\mu,Z)$, defined on another measurable space $(E, \mathcal{E})$, takes values in $L_{0}(\mu,Z)$, the Bochner space of vector-valued functions associated to $\mu$ and $Z$. As a consequence, we also obtain a Rybakov type result for this control. Finally, we give the relation of this result to bounded multiplier properties (BMP) of $F$-spaces and pose various open problems related to it.