Abstract
The classics of function theory (E. Borel, H. Lebesgue, R. Baire, W. H. Young, F. Hausdorff, et al.) have laid down the foundation of the classical descriptive theory of functions. Its initial notions are the notions of a descriptive space and of a measurable function on it. Measurable functions were defined in the classical preimage language. However, a specific range of tasks in theory of functions, measure theory, and integration theory emergent on this base necessitates the usage of the entirely different postclassical cover language, equivalent to the preimage language in the classical case. By means of the cover language, the general notions of a prescriptive space and distributable and uniform functions on it are introduced in this paper and their basic properties are studied.
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References
R. Baire, “Sur les fonctions de variables réelles,” Ann. Mat. Pura Appl. Ser. IIIa, 3, 1–122 (1899).
R. Baire, Leçons sur les fonctions discontinues, Gauthier-Villars, Paris (1905).
E. Borel, Leçons sur la théorie des fonctions, Gauthier-Villars, Paris (1898).
E. Borel, Leçons sur les fonctions de variables réelles, Gauthier-Villars, Paris (1905).
N. T. Fine, L. Gillman, and J. Lambek, Rings of Quotients of Rings of Functions, McGill Univ. Press, Montreal (1965).
F. Hausdorff, “¨Uber halbstetige Funktionen und deren Verallgemeinerung,” Math. Z., 4, 292–309 (1915).
F. Hausdorff, Set Theory, Chelsea Publ., New York (1962).
H. Lebesgue, “Integral, longeur, aire,” Ann. Math. (3), 7 231–359 (1902).
H. Lebesgue, “Sur les séries triginométriques,” Ann. Sci. École Norm. Sup. (3), 20, 453–485 (1903).
H. Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives, Gauthier-Villars, Paris (1904).
A. V. Mikhalev, A. A. Seredinskii, and V. K. Zakharov, “Characterization of the space of Riemann integrable functions by means of cuts of the space of continuous functions. II,” Moscow Univ. Math. Bull., 63, No. 5, 183–192 (2008).
G. Regoli, “Some characterization of sets of measurable functions,” Am. Math. Month., 84, No. 6, 455–458 (1977).
Z. Semadeni, Banach Spaces of Continuous Functions, Mon. Mat., Vol. 55, PWA, Warszawa (1971).
W. Sierpiński, “Sur les fonctions développables en séries absolument convergentes de fonctions continues,” Fund. Math., 2, 15–27 (1921).
M. N. Stone, “Applications of the theory of Boolean rings to general topology,” Trans. Am Math. Soc., 41, 375–481 (1937).
W. H. Young, “On upper and lower integration,” Proc. London Math. Soc. Ser. 2, 2, 52–66 (1905).
W. H. Young, “A new method in the theory of integration,” Proc. London Math. Soc. Ser. 2, 9, 15–50 (1911).
V. K. Zakharov, “Functional characterization of the absolute, vector lattices of functions with the Baire property and of quasinormal functions and modules of particular continuous functions,” Trans. Moscow Math. Soc., No. 1, 69–105 (1984).
V. K. Zakharov, “Alexandrovian cover and Sierpińskian extension,” Stud. Sci. Math. Hungar., 24, No. 2-3, 93–117 (1989).
V. K. Zakharov, “Connections between the Lebesgue extension and the Borel first class extension,
and between the preimages corresponding to them,” Math. USSR Izv., 37, No. 2, 273–302 (1991).
V. K. Zakharov, “Extensions of the ring of continuous functions generated by regular, countably-divisible, complete rings of quotients, and their corresponding pre-images,” Izv. Math., 59, No. 4, 677–720 (1995).
V. K. Zakharov, “Relationships between the Riemann extension and the classical ring of quotients and between the Semadeni preimage and the sequential absolute,” Trans. Moscow Math. Soc., 57, 223–243 (1996).
V. K. Zakharov, “New classes of functions related to general families of sets,” Dokl. Math., 73, No. 2, 197–201 (2006).
V. K. Zakharov, “Hausdorff theorems on measurable functions and a new class of uniform functions,” Moscow Univ. Math. Bull., 63, No. 1, 1–6 (2008).
V. K. Zakharov and A. V. Mikhalev, “Integral representation for Radon measures on an arbitrary Hausdorff space,” Fundam. Prikl. Mat., 3, No. 4, 1135–1172 (1997).
V. K. Zakharov and A. V. Mikhalev, “The problem of general Radon representation for an arbitrary Hausdorff space,” Izv. Math., 63, No. 5, 881–921 (1999).
V. K. Zakharov and A. V. Mikhalev, “Connections between the integral Radon representations for locally compact and Hausdorff spaces,” Fundam. Prikl. Mat., 7, No. 1, 33–46 (2001).
V. K. Zakharov and A. V. Mikhalev, “The problem of the general Radon representation for an arbitrary Hausdorff space. II,” Izv. Math., 66, No. 6, 1087–1101 (2002).
V. K. Zakharov, A. V. Mikhalev, and T. V. Rodionov, “Riesz–Radon–Fréchet problem of characterization of integrals,” Russ. Math. Surv., 65, No. 4, 741–765 (2010).
V. K. Zakharov, A. V. Mikhalev, and T. V. Rodionov, “Characterization of Radon integrals as linear functionals,” J. Math. Sci., 85, No. 2, 233–281 (2012).
V. K. Zakharov and T. V. Rodionov, “A class of uniform functions and its relationship with the class of measurable functions,” Math. Notes, 84, No. 6, 756–770 (2008).
V. K. Zakharov and T. V. Rodionov, “Naturalness of the class of Lebesgue–Borel–Hausdorff measurable functions,” Math. Notes, 95, No. 4, 500–508 (2014).
V. K. Zakharov and A. A. Seredinskii, “A new characterization of Riemann-integrable functions,” J. Math. Sci., 139, No. 4, 6708–6714 (2006).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 6, pp. 77–113, 2014.
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Zakharov, V.K., Mikhalev, A.V. & Rodionov, T.V. Postclassical Families of Functions Proper for Descriptive and Prescriptive Spaces. J Math Sci 221, 360–383 (2017). https://doi.org/10.1007/s10958-017-3231-9
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DOI: https://doi.org/10.1007/s10958-017-3231-9