Lech Drewnowski | Searcharxiv

arXiv2026

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

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.

math.FA↗

arXiv2021

Nonatomic submeasures on $\mathbb N$ and the Banach space $\ell_1$

Nonatomic bounded sequences in $\ell_1$, that is, those giving rise to nonatomic submeasures on $\mathbb N$ are introduced and shown to form a closed subspace nonat$(\ell_1)$ of $\ell_\infty(\ell_1)$, and some spaces of relevant operators in $\ell_1$ are considered.

math.FA↗