Postclassical families of functions proper for descriptive and prescriptive spaces

V. K. Zakharov ; A. V. Mikhalev ; T. V. Rodionov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 6, pp. 77-113

Résumé

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.

Bibliographie

[1] Zakharov V. K., “Funktsionalnaya kharakterizatsiya absolyuta, vektornye reshëtki funktsii so svoistvom Bera i kvazinormalnykh funktsii i moduli chastnykh nepreryvnykh funktsii”, Tr. MMO, 45, no. 1, 1982, 68–104 | MR | Zbl

[2] Zakharov V. K., “Svyazi mezhdu rasshireniem Lebega i rasshireniem Borelya pervogo klassa i mezhdu sootvetstvuyuschimi im proobrazami”, Izv. AN SSSR. Ser. matem., 54:5 (1990), 928–956 | MR | Zbl

[3] Zakharov V. K., “Rasshireniya koltsa nepreryvnykh funktsii, porozhdënnye regulyarnym, schetno-delimym i polnym koltsami chastnykh, i sootvetstvuyuschie im proobrazy”, Izv. RAN. Ser. matem., 59:4 (1995), 15–60 | MR | Zbl

[4] Zakharov V. K., “Svyazi mezhdu rasshireniem Rimana i klassicheskim koltsom chastnykh i mezhdu proobrazom Semadeni i sekventsialnym absolyutom”, Tr. MMO, 57, 1996, 239–262 | MR | Zbl

[5] Zakharov V. K., “Novye klassy funktsii, svyazannye s obschimi semeistvami mnozhestv”, Dokl. RAN, 407:2 (2006), 167–171 | MR | Zbl

[6] Zakharov V. K., “Teoremy Khausdorfa ob izmerimykh funktsiyakh i novyi klass ravnomernykh funktsii”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 63:1 (2008), 3–8 | MR | Zbl

[7] Zakharov V. K., Mikhalëv A. V., “Integralnoe predstavlenie dlya radonovskikh mer na proizvolnom khausdorfovom prostranstve”, Fundament. i prikl. matem., 3:4 (1997), 1135–1172 | MR | Zbl

[8] Zakharov V. K., Mikhalëv A. V., “Problema obschego radonovskogo predstavleniya dlya proizvolnogo khausdorfova prostranstva”, Izv. RAN. Ser. matem., 63:5 (1999), 37–82 | DOI | MR | Zbl

[9] Zakharov V. K., Mikhalëv A. V., “Svyaz mezhdu integralnymi radonovskimi predstavleniyami dlya lokalno kompaktnogo i khausdorfova prostranstv”, Fundament. i prikl. matem., 7:1 (2001), 33–46 | MR | Zbl

[10] Zakharov V. K., Mikhalëv A. V., “Problema obschego radonovskogo predstavleniya dlya proizvolnogo khausdorfova prostranstva. II”, Izv. RAN. Ser. matem., 66:6 (2002), 3–18 | DOI | MR | Zbl

[11] Zakharov V. K., Mikhalëv A. V., Rodionov T. V., “Problema Rissa–Radona–Freshe kharakterizatsii integralov”, UMN, 65:4 (2010), 153–178 | DOI | MR | Zbl

[12] Zakharov V. K., Mikhalëv A. V., Rodionov T. V., “Opisanie radonovskikh integralov kak lineinykh funktsionalov”, Fundament. i prikl. matem., 16:8 (2010), 87–161 | MR

[13] Zakharov V. K., Mikhalëv A. V., Seredinskii A. A., “Kharakterizatsiya prostranstva funktsii, integriruemykh po Rimanu, posredstvom sechenii prostranstva nepreryvnykh funktsii. II”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 63:5 (2008), 11–20 | MR | Zbl

[14] Zakharov V. K., Rodionov T. V., “Klass ravnomernykh funktsii i ego sootnoshenie s klassom izmerimykh funktsii”, Matem. zametki, 84:6 (2008), 809–824 | DOI | MR | Zbl

[15] Zakharov V. K., Rodionov T. V., “Estestvennost klassa izmerimykh funktsii v smysle Lebega–Borelya–Khausdorfa”, Matem. zametki, 95:4 (2014), 554–563 | DOI | MR | Zbl

[16] Zakharov V. K., Seredinskii A. A., “Novaya kharakterizatsiya funktsii, integriruemykh po Rimanu”, Fundament. i prikl. mat., 10:3 (2004), 73–83 | MR | Zbl

[17] Khausdorf F., Teoriya mnozhestv, URSS, M., 2004

[18] Baire R., “Sur les fonctions de variables réelles”, Ann. Mat. Pura Appl. Ser. IIIa, 3 (1899), 1–122 | DOI

[19] Baire R., Leçons sur les fonctions discontinues, Gauthier-Villars, Paris, 1905 | Zbl

[20] Borel E., Leçons sur la théorie des fonctions, Gauthier-Villars, Paris, 1898

[21] Borel E., Leçons sur les fonctions de variables réelles, Gauthier-Villars, Paris, 1905 | Zbl

[22] Fine N. T., Gillman l., Lambek J., Rings of Quotients of Rings of Functions, McGill Univ. Press, Montreal, 1965 | MR

[23] Hausdorff F., “Über halbstetige Funktionen und deren Verallgemeinerung”, Math. Z., 4 (1915), 292–309 | MR

[24] Lebesgue H., “Integral, longeur, aire”, Ann. Math. (3), 7 (1902), 231–359 | Zbl

[25] Lebesgue H., “Sur les séries triginométriques”, Ann. Sci. École Norm. Sup. (3), 20 (1903), 453–485 | MR | Zbl

[26] Lebesgue H., Leçons sur l'integration et la recherche des fonctions primitives, Gauthier-Villars, Paris, 1904

[27] Regoli G., “Some characterization of sets of measurable functions”, Am. Math. Month., 84:6 (1977), 455–458 | DOI | MR | Zbl

[28] Semadeni Z., Banach Spaces of Continuous Functions, Mon. Mat., 55, PWA, Warszawa, 1971 | MR

[29] Sierpiński W., “Sur les fonctions développables en séries absolument convergentes de fonctions continues”, Fund. Math., 2 (1921), 15–27 | Zbl

[30] Stone M. N., “Applications of the theory of Boolean rings to general topology”, Trans. Am. Math. Soc., 41 (1937), 375–481 | DOI | MR | Zbl

[31] Young W. H., “On upper and lower integration”, Proc. London Math. Soc. Ser. 2, 2 (1905), 52–66 | DOI | MR

[32] Young W. H., “A new method in the theory of integration”, Proc. London Math. Soc. Ser. 2, 9 (1911), 15–50 | DOI | MR

[33] Zakharov V. K., “Alexandrovian cover and Sierpińskian extension”, Stud. Sci. Math. Hungar., 24:2–3 (1989), 93–117 | MR | Zbl