Abstract
A result with numerous applications in economics is the theorem of Lyapunov (1940), which states that the range of a nonatomic totally finite vector-valued measure is both convex and compact. That is, let Σ be a σ-algebra in a set X and let μ1, μ2,…,μk be totally finite, nonatomic, signed measures on the measurable space (X, Σ). Then the range of the vector measure \( \overline{\mu}=\left({\mu}_1,{\mu}_2\dots {\mu}_k\right) \), that is, the set \( \left\{\overline{\mu}(A):A\in \sum \right\} \), is a convex, compact subset of Rk.
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Loeb, P.A., Rashid, S. (2018). Lyapunov’s Theorem. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95189-5_1175
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DOI: https://doi.org/10.1057/978-1-349-95189-5_1175
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