Control Measures for Bochner $L_0$-Valued Vector Measures

Abstract

It is shown that for any finite positive measure $\hat{I}¼$ defined on a measure space $(S,\hat{I}\pounds )$, and any Banach or Fréchet space $Z$, the control measure Theorem of Talagrand (T) is true for the case when the (stochastic) vector measure $\mathbf{m}:\mathcal{E}\to L_0(\hat{I}¼,Z)$, defined on another measurable space $(E,\mathcal{E})$, takes values in $L_0(\hat{I}¼,Z)$, the Bochner space of vector-valued functions associated to $\hat{I}¼$ 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.