Abstract. For a non-negative finite countably additive measure µ defined on the σ-field Σ of subsets of Ω, it is well known that a certain quotient of Σ can be turned into a complete metric space Σ(Ω), known as the Nikodym-Saks space, which yields such important results in Measure Theory and Functional Analysis as Vitali-Hahn-Saks and Nikodym's theorems. Here we study some topological properties of Σ(Ω) regarded as a quasi-pseudometric space.2000 AMS Classification: 54E15, 54E55.
Liapounoff, in 1940, proved that the range of a countably additive bounded measure with values in a finite dimensional vector space is compact and, in the nonatomic case, is convex. Later, in 1945, Liapounoff showed, by counterexample, that neither the convexity nor compactness need hold in the infinite dimensional case. The next step was taken by Halmos who in 1948 gave simplified proofs of Liapounoff's results for the finite dimensional case. In 1951, Blackwell [l] considered the case of a measure represented by a finite dimensional vector integral and obtained results similar to those of Liapounoff for these measures. Various versions of Liapounoff's theorem
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