The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Lyapunov’s Theorem

  • Peter A. Loeb
  • Salim Rashid
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_1175

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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Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Peter A. Loeb
    • 1
  • Salim Rashid
    • 1
  1. 1.