Kontolatou, Angeliki Measure and measurable functions of \(S^ n\). (English) Zbl 0876.28004 Acta Math. Inform. Univ. Ostrav. 2, No. 1, 85-99 (1994). Summary: Let \(\mathbb{R}\) be the set of real numbers ordered by the usual ordering, \(\widehat{\mathbb{R}}= \mathbb{R}\cup\{-\infty,+\infty\}\) and \(\Xi=\{-,0,+\}\) with \(-<0<+\). The set \(S=\widehat{\mathbb{R}}\times\Xi\backslash \{(-\infty,-),(+\infty,+)\}\) ordered lexicographically and endowed by some partial operations and the order topology, is said to be the quasi-real line and its elements the quasi-real numbers. We clear the disconnected character of \(S\), we give a measure on \(S^n\) and generalize an extension theorem on real-valued functions ranging over a more general than \(S^n\) partially ordered set. The last theorem shows that such a function, under easy conditions, is extended into a continuous function. MSC: 28A12 Contents, measures, outer measures, capacities 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 28E99 Miscellaneous topics in measure theory 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence Keywords:hypercube; measures; measurable functions; quasi-real numbers; extension; real-valued functions; partially ordered set × Cite Format Result Cite Review PDF Full Text: EuDML References: [1] Birkhoff G.: Lattice Theory. AMS, Providence, Rhode Island, 1967. · Zbl 0153.02501 [2] Dixmier J.: Cours de Mathematiques. (Cahiers scientifiques) Gauthier Villars, Paris, 1968. · Zbl 0162.07101 [3] Dokas L.: Analyse réele. Classes de Baire des fonctions réeles d’une variable semi-réele. C.R. Acad. Sc. Paris, p. 1835-1837, 1965. · Zbl 0127.02603 [4] Dokas L.: Sur les classes de Baire des fonctions semi-réelles. Bull. de la Soc. Mathematique de Grece 15 (1974), 83-108. · Zbl 0309.26006 [5] Halmos P.: Measure Theory. The University series, 6, Van Nostrand Prinseton, New York, London, 1966. [6] Kontolatou A.: The Quasi-real extension of the real numbers. Acta Mathemat. et Inform., Univ. Ostr. 1 (1993), 27-36. · Zbl 0868.13020 [7] Krasner M.: Nombres semi-réeles, et espaces ultrametriques. C. R. Acad. Sc. Paris 219 (1944), 433-435. · Zbl 0061.06202 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.