Real Analysis Exchange

Essential Closures

Pongpol Ruankong and Songkiat Sumetkijakan

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Based on the Zermelo-Fraenkel system of axioms ZF, we introduce a theory of essential closures. It is a generalization of the concept of topological closures. A typical essential closure collects all points which are essential with respect to a submeasure; hence it is called a submeasure closure. One of our main results states that a “nice” essential closure must be a submeasure closure. Many examples of known and new submeasure closures are discussed and their applications are demonstrated, especially in the study of the supports of measures.

Article information

Real Anal. Exchange, Volume 41, Number 1 (2016), 55-86.

First available in Project Euclid: 29 March 2017

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 28A35: Measures and integrals in product spaces
Secondary: 46B20: Geometry and structure of normed linear spaces 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx} 60B10: Convergence of probability measures

topological closures essential closures submeasures non-essential sets lower density operators stochastic closures


Ruankong, Pongpol; Sumetkijakan, Songkiat. Essential Closures. Real Anal. Exchange 41 (2016), no. 1, 55--86.

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