Geodesic
On a theorem of Dvoretsky, Wald, and Wolfowitz concerning Liapounov Measures
Glasgow mathematical journal, Tome 29 (1987) no. 2, pp. 205-220

Voir la notice de l'article provenant de la source Cambridge University Press

Let ω be a non-empty set, F a Boolean σ-algebra of subsets of Ω, k a natural number, and let m:F→Rk be a non-atomic vector measure. Then, by the celebrated theorem of Liapounov [11], the range m[3F] = {m(A): A ε F3F} of m is a compact convex subset of Rk. 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 vectorswhere (A1 A2,..., An) is an ordered F-measurable partition of Ω (i.e. a partition whose terms A, all belong to F). They prove in [6] that Kn is a compact convex subset of Rnk and moreover that Kn is equal to the set of all vectors of the formwhere (φ1, φ2..., φn) is an F-measurable partition of unity; i.e. it is an n-tuple of non-negative φr on Ω such thatLiapounov's theorem can be obtained as a corollary of this result by taking n= 2. 
Edwards, D. A. On a theorem of Dvoretsky, Wald, and Wolfowitz concerning Liapounov Measures. Glasgow mathematical journal, Tome 29 (1987) no. 2, pp. 205-220. doi: 10.1017/S0017089500006856
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