On the range of an unbounded partly atomic vector-valued measure (Q2644792)

A well-known theorem of Lyapunov states that the range of a bounded, non- atomic, finite-dimensional vector-valued measure is closed and convex. In this paper we study the range of an unbounded finite-dimensional vector- valued measure that is at least partly atomic. In the one-dimensional case we show that if the range is dense in an interval [0,a] for some \(a>0\) then it contains [0,a]. In the general case of arbitrary dimension d we shall use the following notation. If \(e_ 1,...,e_ d\) are linearly independent vectors in \({\mathbb{R}}^ d\) let \(C^ 0\) denote the interior of the convex cone \(C=\{a_ 1e_ 1+...+a_ de_ d:\;a_ 1,...,a_ d\geq 0\}.\) Then, if \(x=a_ 1e_ 1+...+a_ de_ d\) and \(y=b_ 1e_ 1+...+b_ de_ d\) are in C, \(x<y\) and \(x\leq y\) shall mean that \(a_ k<b_ k\) and that \(a_ k\leq b_ k\), respectively, for \(k=1,...,d\). Finally, if \(a\in C^ 0\) define \((0,a)=\{x\in C^ 0:\;0<x<a\}\) and \((0,a]=\{x\in C^ 0:\;0<x\leq a\}.\) Now let \(\mu\) be a measure such that any bounded subset of its range is in C, and such that the set of all \(\mu\) (E) in C such that E contains no atom is bounded. We show that if its range is dense in (0,a] for some \(a\in C^ 0\) then it contains (0,a].