Glasgow Mathematical Journal

Research Article

On a theorem of Dvoretsky, Wald, and Wolfowitz concerning Liapounov Measures

D. A. Edwardsa1

a1 Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB.

Let ω be a non-empty set, xs2131 a Boolean σ-algebra of subsets of Ω, k a natural number, and let m:xs2131xs211Dk be a non-atomic vector measure. Then, by the celebrated theorem of Liapounov [11], the range m[3F] = {m(A): A ε xs21313F} of m is a compact convex subset of xs211Dk. This theorem has been generalized in a number of ways. For example Kingman and Robertson [8] and Knowles [9] have shown that, under appropriate conditions, results in the same spirit can be proved for measures taking their values in infinite-dimensional vector spaces. Another type of generalization was obtained by Dvoretsky, Wald and Wolfowitz [6,7]. What they do is to take m as above together with a natural number n≥ 1. They then consider the set Knof all vectors

S0017089500006856_eqnU1

where (A1 A2,…, An) is an ordered xs2131-measurable partition of Ω (i.e. a partition whose terms A, all belong to xs2131). They prove in [6] that Kn is a compact convex subset of xs211Dnk and moreover that Kn is equal to the set of all vectors of the form

S0017089500006856_eqnU2

where (xs03D51, xs03D52…, xs03D5n) is an xs2131-measurable partition of unity; i.e. it is an n-tuple of non-negative xs03D5r on Ω such that

S0017089500006856_eqnU3

Liapounov's theorem can be obtained as a corollary of this result by taking n= 2.

(Received January 25 1986)