On Multivalued Amarts

Tom 52 / 2004

Dorota Dudek, Wiesław Zięba Bulletin Polish Acad. Sci. Math. 52 (2004), 93-99 MSC: 60G48, 60F15, 28B. DOI: 10.4064/ba52-1-10


In recent years, convergence results for multivalued functions have been developed and used in several areas of applied mathematics: mathematical economics, optimal control, mechanics, etc. The aim of this note is to give a criterion of almost sure convergence for multivalued asymptotic martingales (amarts). For every separable Banach space $B$ the fact that every $L^{1}$-bounded $B$-valued martingale converges a.s. in norm to an integrable $B$-valued random variable (r.v.) is equivalent to the Radon–Nikodym property [6]. In this paper we solve the problem of a.s. convergence of multivalued amarts by giving a topological characterization.


  • Dorota DudekInstitute of Mathematics
    Maria Curie-Skłodowska University
    Pl. Marii Curie-Skłodowskiej 1
    20-031 Lublin, Poland
  • Wiesław ZiębaInstitute of Mathematics
    Maria Curie-Skłodowska University
    Pl. Marii Curie-Słodowskiej 1
    20-031 Lublin, Poland

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