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A well-known theorem of
Liapunov 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 e1,…,ed are linearly independent vectors in ℝd let
C∘ denote the interior of the convex cone C = {a1e1 + ⋯ + aded : a1,…,ad ≥ 0}. Then,
if x = a1e1 + ⋯ + aded and y = b1e1 + ⋯ + bded are in C, x < y and x ≤ y shall mean
that ak < bk and that ak ≤ bk, respectively, for k = 1,…,d. Finally, if a ∈ C∘
define (0,a) = {x ∈ C∘ : 0 < x < a} and (0,a] = {x ∈ C∘ : 0 < x ≤ a}.
Now let μ be a measure such that any bounded subset of its range is in
C, and such that the set of all μ(E) in C such that E contains no atom is
bounded. We show that if Rμ is dense in (0,a] for some a ∈ C∘ then it contains
(0,a].
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