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Keywords:
Henstock-Kurzweil integral; charge; $\rm BV$ set
Henstock-Kurzweil integral; charge; $\rm BV$ set
Summary:
The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $\rm BV$ subsets of $\mathbb R^n$, but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $\sigma $-finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $\rm BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect to the formation of improper integrals. Its definition in $\mathbb R$ coincides with the Henstock-Kurzweil definition of the Denjoy-Perron integral.
References:
[1] Denjoy, A.: Une extension de l'intégrale de M. Lebesgue. C. R. Acad. Sci., Paris 154 (1912), 859-862 French.
[2] Pauw, T. De, Pfeffer, W.F.: Distributions for which $ div v={F}$ has continuous solution. Commun. Pure Appl. Math. 61 (2008), 230-260. DOI 10.1002/cpa.20204 | MR 2368375
[3] Gordon, R.A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics 4 American Mathematical Society, Providence (1994). DOI 10.1090/gsm/004/09 | MR 1288751 | Zbl 0807.26004
[4] Henstock, R.: A Riemann type integral of Lebesgue power. Can. J. Math. 20 (1968), 79-87. DOI 10.4153/CJM-1968-010-5 | MR 0219675 | Zbl 0171.01804
[5] Henstock, R.: Definitions of Riemann type of the variational integrals. Proc. Lond. Math. Soc. (3) 11 (1961), 402-418. DOI 10.1112/plms/s3-11.1.402 | MR 0132147 | Zbl 0099.27402
[6] Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7(82) (1957), 418-449. MR 0111875 | Zbl 0090.30002
[7] Jarník, J., Kurzweil, J.: A non absolutely convergent integral which admits transformation and can be used for integration on manifolds. Czech. Math. J. 35(110) (1985), 116-139. MR 0779340 | Zbl 0614.26007
[8] Jarník, J., Kurzweil, J.: A nonabsolutely convergent integral which admits $C^1$-transfor-mations. Čas. Pěst. Mat. 109 (1984), 157-167. MR 0744873
[9] Jarník, J., Kurzweil, J., Schwabik, Š.: On Mawhin's approach to multiple nonabsolutely convergent integral. Čas. Pěst. Mat. 108 (1983), 356-380. MR 0727536 | Zbl 0555.26004
[10] Mawhin, J.: Generalized multiple Perron integrals and the Green-{G}oursat theorem for differentiable vector fields. Czech. Math. J. 31(106) (1981), 614-632. MR 0631606 | Zbl 0562.26004
[11] 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 Symp. in Honour of Christoffel on the 150th Anniversary of his Birth, Aachen and Monschau, Germany, 1979 Birkhäuser, Basel (1981), 704-714 P. L. Butzer et al. MR 0661109 | Zbl 0562.26003
[12] Perron, O.: Über den Integralbegriff. Heidelb. Ak. Sitzungsber. 14 Heidelberg German (1914).
[13] Pfeffer, W.F.: The Divergence Theorem and Sets of Finite Perimeter. Pure and Applied Mathematics 303 CRC Press, Boca Raton (2012). DOI 10.1201/b11919-10 | MR 2963550 | Zbl 1258.28002
[14] Pfeffer, W.F.: Derivatives and primitives. Sci. Math. Jpn. 55 (2002), 399-425. MR 1887074 | Zbl 1010.26012
[15] Pfeffer, W.F.: Derivation and Integration. Cambridge Tracts in Mathematics 140 Cambridge University Press, Cambridge (2001). MR 1816996 | Zbl 0980.26008
[16] Pfeffer, W.F.: The {Gauss-Green} theorem. Adv. Math. 87 (1991), 93-147. DOI 10.1016/0001-8708(91)90063-D | MR 1102966 | Zbl 0732.26013
[17] Pfeffer, W.F.: The multidimensional fundamental theorem of calculus. J. Aust. Math. Soc., Ser. A 43 (1987), 143-170. DOI 10.1017/S1446788700029293 | MR 0896622 | Zbl 0638.26011
[18] Pfeffer, W.F.: A Riemann-type integral and the divergence theorem. C.R. Acad. Sci., Paris, (1) 299 (1984), 299-301 French. MR 0761251 | Zbl 0574.26009
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