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On a generalization of Henstock-Kurzweil integrals
Mathematica Bohemica, Tome 144 (2019) no. 4, pp. 393-422
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We study a scale of integrals on the real line motivated by the $MC_{\alpha }$ integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.
We study a scale of integrals on the real line motivated by the $MC_{\alpha }$ integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.
DOI : 10.21136/MB.2019.0038-19
Classification : 26A39
Keywords: Henstock-Kurzweil integral 
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Malý, Jan; Kuncová, Kristýna. On a generalization of Henstock-Kurzweil integrals. Mathematica Bohemica, Tome 144 (2019) no. 4, pp. 393-422. doi: 10.21136/MB.2019.0038-19

[1] Ball, T., Preiss, D.: Monotonically controlled integrals. Mathematics Almost Everywhere. In Memory of Solomon Marcus World Scientific Publishing, Hackensack A. Bellow et al. (2018), 69-92. | DOI | MR

[2] Bendová, H., Malý, J.: An elementary way to introduce a Perron-like integral. Ann. Acad. Sci. Fenn., Math. 36 (2011), 153-164. | DOI | MR | JFM

[3] Bongiorno, B.: The Henstock-Kurzweil integral. Handbook of Measure Theory. Vol. I and II North-Holland, Amsterdam (2002), 587-615 E. Pap. | DOI | MR | JFM

[4] Burkill, J. C.: The approximately continuous Perron integral. Math. Z. 34 (1932), 270-278. | DOI | MR | JFM

[5] Pauw, T. De, Pfeffer, W. F.: Distributions for which $ div v=F$ has a continuous solution. Commun. Pure Appl. Math. 61 (2008), 230-260. | DOI | MR | JFM

[6] Denjoy, A.: Une extension de l'intégrale de M. Lebesgue. C. R. Acad. Sci., Paris Sér. I Math. 154 (1912), 859-862 French. | JFM

[7] Denjoy, A.: Sur la dérivation et son calcul inverse. C. R. Acad. Sci., Paris Sér. I Math. 162 (1916), 377-380 French. | MR | JFM

[8] Evans, L. C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992). | MR | JFM

[9] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics 4. AMS, Providence (1994). | DOI | MR | JFM

[10] Gordon, R. A.: Some comments on an approximately continuous Khintchine integral. Real Anal. Exch. 20 (1994-95), 831-841. | DOI | MR | JFM

[11] Hencl, S.: On the notions of absolute continuity for functions of several variables. Fundam. Math. 173 (2002), 175-189. | DOI | MR | JFM

[12] Henstock, R.: Definitions of Riemann type of the variational integrals. Proc. London Math. Soc., III. Ser. 11 (1961), 402-418. | DOI | MR | JFM

[13] Henstock, R.: Theory of Integration. Butterworths Mathematical Texts. Butterworths & Co., London (1963). | MR | JFM

[14] Henstock, R.: The General Theory of Integration. Oxford Mathematical Monographs. Clarendon Press, Oxford (1991). | MR | JFM

[15] Honzík, P., Malý, J.: Non-absolutely convergent integrals and singular integrals. Collect. Math. 65 (2014), 367-377. | DOI | MR | JFM

[16] Jarník, J., Kurzweil, J.: A non-absolutely convergent integral which admits $C^1$-transformations. Čas. Pěst. Mat. 109 (1984), 157-167. | MR | JFM

[17] Jarník, J., Kurzweil, J.: A non absolutely convergent integral which admits transformation and can be used for integration on manifolds. Czech. Math. J. 35 (1985), 116-139. | DOI | MR | JFM

[18] Jarník, J., Kurzweil, J.: A new and more powerful concept of the PU-integral. Czech. Math. J. 38 (1988), 8-48. | DOI | MR | JFM

[19] Jarník, J., Kurzweil, J., Schwabik, Š.: On Mawhin's approach to multiple nonabsolutely convergent integral. Čas. Pěst. Mat. 108 (1983), 356-380. | MR | JFM

[20] Jurkat, W. B.: The divergence theorem and Perron integration with exceptional sets. Czech. Math. J. 43 (1993), 27-45. | DOI | MR | JFM

[21] Khintchine, A.: Sur une extension de l'intégrale de M. Denjoy. C. R. Acad. Sci. Paris, Sér. I Math. 162 (1916), 287-291 French. | JFM

[22] Kubota, Y.: An integral of the Denjoy type. Proc. Japan Acad. 40 (1964), 713-717. | DOI | MR | JFM

[23] Kuncová, K.: $BV$-packing integral in $\Bbb R^n$. \ Available at (2019). | arXiv | MR

[24] Kuncová, K., Malý, J.: Non-absolutely convergent integrals in metric spaces. J. Math. Anal. Appl. 401 (2013), 578-600. | DOI | MR | JFM

[25] Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7 (1957), 418-449. | DOI | MR | JFM

[26] Kurzweil, J.: Nichtabsolut konvergente Integrale. Teubner-Texte zur Mathematik, Bd. 26. BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1980), German. | DOI | MR | JFM

[27] Lusin, N.: Sur les propriétés de l'intégrale de M. Denjoy. C. R. Acad. Sci. Paris Sér. I Math. 155 (1912), 1475-1477 French. | MR | JFM

[28] Malý, J.: Non-absolutely convergent integrals with respect to distributions. Ann. Mat. Pura Appl. (4) 193 (2014), 1457-1484. | DOI | MR | JFM

[29] Malý, J., Pfeffer, W. F.: Henstock-Kurzweil integral on BV sets. Math. Bohem. 141 (2016), 217-237. | DOI | MR | JFM

[30] Mawhin, J.: Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields. Czech. Math. J. 31 (1981), 614-632. | DOI | MR | JFM

[31] Mawhin, J.: Generalized Riemann integrals and the divergence theorem for differentiable vector fields. E. B. Christoffel: The Influence of his Work on Mathematics and The Physical Sciences. Int. Christoffel Sym., Aachen and Monschau, 1979 Birkhäuser, Basel (1981), 704-714 P. L. Butzer et al. | DOI | MR | JFM

[32] Monteiro, G. A., Slavík, A., Tvrdý, M.: Kurzweil-Stieltjes Integral. Theory and Applications. Series in Real Analysis 15. World Scientific Publishing, Hackensack (2018). | DOI | MR | JFM

[33] Novikov, A., Pfeffer, W. F.: An invariant Riemann type integral defined by figures. Proc. Am. Math. Soc. 120 (1994), 849-853. | DOI | MR | JFM

[34] Perron, O.: Über den Integralbegriff. Heidelb. Ak. Sitzungsber 14 (1914), 1-16 German. | JFM

[35] Pfeffer, W. F.: Une intégrale Riemannienne et le théorème de divergence. C. R. Acad. Sci. Paris, Sér. I Math. 299 (1984), 299-301 French. | MR | JFM

[36] Pfeffer, W. F.: The multidimensional fundamental theorem of calculus. J. Aust. Math. Soc., Ser. A 43 (1987), 143-170. | DOI | MR | JFM

[37] Pfeffer, W. F.: The Gauss-Green theorem. Adv. Math. 87 (1991), 93-147. | DOI | MR | JFM

[38] Pfeffer, W. F.: Derivation and Integration. Cambridge Tracts in Mathematics 140. Cambridge University Press, Cambridge (2001). | DOI | MR | JFM

[39] Pfeffer, W. F.: Derivatives and primitives. Sci. Math. Jpn. 55 (2002), 399-425. | MR | JFM

[40] Preiss, D., Thomson, B. S.: The approximate symmetric integral. Can. J. Math. 41 (1989), 508-555. | DOI | MR | JFM

[41] Saks, S.: Theory of the Integral. With two Additional Notes by Stefan Banach. Dover Publications, Mineola (1964). | MR | JFM

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