On the range of an unbounded vector-valued measure

1. In t roduc t ion . Consider a vector-valued measure (S, E, /~), that is, a space S, a o'-field E o f subsets o f S and a countably additive funct ion def ined on E and taking values f rom R n if finite, or infinity deno ted by o0. We assume that 0¢ + a = oo, E~=I a, = 0e if II~gol a,II as n ~ o0. T h e purpose o f this note is to describe the range o f such a measure. In the case that /z(S) is finite and the m e a s u r e / z is non-atomic, then a result due to A. A. Liapunov [4] says that the range o f / z is compact and convex. In the case we consider, the range remains convex, as can be easily seen f rom the Liapunov theorem, but need not be closed. However , we have the following result.