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Connecting two points in the range of a vector measure

Volume 153 / 2018

Jerzy Legut Colloquium Mathematicum 153 (2018), 163-167 MSC: Primary 28B05; Secondary 52A20. DOI: 10.4064/cm7176-9-2017 Published online: 27 April 2018

Abstract

Let $\mu =( \mu _{1},\ldots ,\mu _{n}) $ be a nonatomic vector measure defined on measurable subsets $ \cal {B} $ of the unit interval $I= [0,1] $. Denote by $ {\cal U}(k) $ the collection of all sets that are unions of at most $ k $ disjoint subintervals of $ I $. We show that if $ A \in {\cal U}(k) $ then the line segment connecting $0\in \mathbb {R}^n$ and $ \mu (A) $ is contained in $ \mu ( {\cal U}(n+k-1))$. Moreover, if $B,C \in {\cal U}(k) $ then the line segment connecting $\mu (B)$ and $ \mu (C)$ is a subset of $ \mu ( {\cal U}(2n+4k-3))$. This result is used to give yet another proof of the Lyapunov convexity theorem. We also discuss the two-dimensional case for specific vector measures.

Authors

  • Jerzy LegutFaculty of Pure and Applied Mathematics
    Wrocław University of Science and Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail

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