Borel multivalued mappings

About the author:

Anton P. Devyatkov, Cand. Sci. (Phys.-Math.), Associate Professor, Department of Mathematical Analysis and Functions Theory, Institute of Mathematics and Computer Sciences, Tyumen State University

Abstract:

The article studies multivalued mappings of metric space Y into compact metric space X. It is demonstrated that semi continuous multivalued mappings are Borel mappings of the first class. The author investigates whether Borel measurability remains after the operations of intersection, union, taking upper and lower topological limit performed on multi-valued mappings. It is demonstrated that the operations of intersection of finite or countable number of mappings, and also operation of union of countable number of mappings increase a Borel class by one; the operation of union of finite number of mappings does not change a Borel class; the operation of tacking upper topological limit of sequence of multi-valued mappings increases a Borel class by two; the operation of tacking lower topological limit increases a Borel class by three. These results are applied further to the mappings determined by radial limit sets.

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