On control measures of multimeasures

TL;DR

This paper investigates conditions under which multimeasures in Banach spaces admit control measures, establishing that the absence of a specific subspace is crucial for such measures to exist.

Contribution

It proves that multimeasures have control measures if and only if the Banach space does not contain a subspace isomorphic to c_0(ω_1), highlighting the importance of this geometric condition.

Findings

Multimeasures admit control measures under certain space conditions.

The absence of c_0(ω_1) in the Banach space is necessary.

The additional assumption on the space is shown to be essential.

Abstract

Let $M$ be a multimeasure defined on a $\sigma$-algebra and taking values in the family of bounded non-empty subsets of a Banach space $X$. We prove that $M$ admits a control measure whenever $X$ contains no subspace isomorphic to $c_0(\omega_1)$. The additional assumption on $X$ is shown to be essential.