On Atomicity of External Measures

УДК 515.123

  • A.N. Sazhenkov Altai State University (Barnaul, Russia) Email: sazhenkov_an@mail.ru
  • E.A. Plotnikova Novosibirsk State Technical University (Novosibirsk, Russia) Email: pselena@gmail.com
DOI_disc_logo https://doi.org/10.14258/izvasu(2019)4-17
Keywords: exterior measure, continuity, boundedness, topology, atomicity of measure

Abstract

We consider the notion of non-atomicity for abstract exterior measures being defined on the Boolean ring and taking values in an arbitrary set. For scalar functions of sets, non-atomicity of a set means that it can be represented as a finite union of pairwise non-intersecting sets with the following property: for any belonging to the ring subset of any of these non-intersecting sets, absolute value of the scalar function applied to the subset is less than any afore given positive value.

In this work, associated with the exterior measure topologies are consistent with the structure of the Boolean ring and have the specially constructed filter base of neighborhoods of zero. In this topology of operations, conjunction, disjunction, addition, and multiplication operators are uniformly continuous. Survey of nowadays state of study of topics related to exterior measures and topologies can be found in the works of L. Y. Saveliev.

We focus on the notion of abstract exterior measure, which covers a wide class of measures. Also, we consider the notion of exterior topology. We set two definitions of boundedness in topological Abelian group and investigate interconnection of these definitions. In the main results of the work, the notion of nonatomicity of measure is characterized as the property of boundedness of the domain of definition of exterior measure in an appropriate topological group.

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Author Biographies

A.N. Sazhenkov, Altai State University (Barnaul, Russia)

кандидат физико-математических наук, доцент кафедры математического анализа

E.A. Plotnikova, Novosibirsk State Technical University (Novosibirsk, Russia)

кандидат физико-математических наук, доцент кафедры высшей математики

References

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Published
2019-09-12
How to Cite
Sazhenkov A., Plotnikova E. On Atomicity of External Measures // Izvestiya of Altai State University, 2019, № 4(108). P. 104-107 DOI: 10.14258/izvasu(2019)4-17. URL: http://izvestiya.asu.ru/article/view/%282019%294-17.
Issue
No 4(108) (2019): Izvestiya of Altai State University
Section
Математика и механика

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