Measurable multifunction theorems in different spaces and their applications

Abstract

In recent years, significant research efforts have been devoted to exploring measurable multifunctions along with their basic characteristics in different contexts within mathematics. In particular, Polish spaces, separable Banach spaces, and even non-separable Banach spaces have been considered as domains for multifunctions. This paper provides a comprehensive discussion on measurable multifunctions in terms of notions like measurability, measurable selections, graph measurability, and representations of set-valued mappings. The classical selection theorem by Kuratowski and Ryll- Nardzewski and the representation theorem by Castaing have been analyzed extensively. Additionally, the concept of graph measurability has been extended to the case of non-separable Banach spaces. The concept of Aumann integral as an extension of Lebesgue integral to multifunctions has been defined and some important features of Aumann integrals have been outlined. Finally, the relevance of measurable multifunction theory to fixed point theory, differential inclusion problems, optimization issues, and stochastic calculus has been illustrated.