# Article

Full entry | PDF   (0.3 MB)
Keywords:
absolute continuity; quasi-uniformity; acceptable mapping
Summary:
The aim of this paper is to introduce a generalization of the classical absolute continuity to a relative case, with respect to a subset \$M\$ of an interval \$I\$. This generalization is based on adding more requirements to disjoint systems \$\{(a_k, b_k)\}_K\$ from the classical definition of absolute continuity -- these systems should be not too far from \$M\$ and should be small relative to some covers of \$M\$. We discuss basic properties of relative absolutely continuous functions and compare this class with other classes of generalized absolutely continuous functions.
References:
[1] Bartle R.G.: Modern Theory of Integration. American Mathematical Society, Providence, RI, 2001. MR 1817647 | Zbl 0968.26001
[2] Bogachev V.I.: Measure Theory I. Springer, Berlin, Heidelberg, 2007. MR 2267655
[3] Ene V.: Characterisations of VBG \$\cap\$ (N). Real Anal. Exch. 23 (1997-1998), no. 2, 611–630. MR 1639992
[4] Ene V.: Characterisations of VB\$^{\ast}\$G \$\cap\$ (N). Real Anal. Exch. 23 (1997-1998), no. 2, 571–600. MR 1639984
[5] Gong Z.: New descriptive characterisation of Henstock-Kurzweil integral. Southeast Asian Bull. Math., 2003, no. 27, 445–450. MR 2045557
[6] Gordon R.A.: A descriptive characterization of the generalized Riemann integral. Real Anal. Exch. 15 (1990), no. 1, 397–400. MR 1042557 | Zbl 0703.26009
[7] Gordon R.A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. American Mathematical Society, Providence, RI, 1994. MR 1288751 | Zbl 0807.26004
[8] Fletcher P., Lindgren W.F.: Quasi-uniform Spaces. Lecture Notes in Pure and Applied Mathematics, 77, Dekker, New York, 1982. MR 0660063 | Zbl 0583.54017
[9] Isbell J.R.: Uniform Spaces. American Mathematical Society, Providence, RI, 1964. MR 0170323 | Zbl 0124.15601
[10] Lee P.Y.: On \$ {ACG}^{\ast}\$ functions. Real Anal. Exch. 15 (1990), no. 2, 754–759. MR 1059436
[11] Lee P.Y.: Lanzhou Lectures on Henstock Integration. World Scientific, Singapore, 1989. MR 1050957 | Zbl 0699.26004
[12] Njastad O.: On uniform spaces where all uniformly continuous functions are bounded. Monatsh. Math. 69 (1965), no. 2, 167–176. DOI 10.1007/BF01298321 | MR 0178453 | Zbl 0145.19502
[13] Saks S.: Theory of the Integral. Warsawa, Lwów, 1937. Zbl 0017.30004
[14] Sworowski P.: On the uniform strong Lusin condition. Math. Slovaca 63 (2013), no. 2, 229-–242. DOI 10.2478/s12175-012-0095-9 | MR 3037065 | Zbl 1324.26010
[15] Zhereby'ev Yu.A.: On the Denjoy-Luzin definitions of the function classes \$ACG\$, \$ACG^{\ast}\$, \$VBG\$, and \$VBG^{\ast}\$. Mathematical Notes 81 (2007), no. 2, 183–192. DOI 10.1134/S000143460701021X | MR 2409272

Partner of