Previous |  Up |  Next

Article

Keywords:
approximate Lusin function; ap-Denjoy integral; ap-Henstock integral; choice
Summary:
In this paper we define the ap-Denjoy integral and show that the ap-Denjoy integral is equivalent to the ap-Henstock integral and the integrals are equal.
References:
[1] P. S.  Bullen: The Burkill approximately continuous integral. J.  Austral. Math. Soc. (Ser.  A) 35 (1983), 236–253. DOI 10.1017/S1446788700025738 | MR 0704431 | Zbl 0533.26006
[2] T. S.  Chew, K.  Liao: The descriptive definitions and properties of the AP-integral and their application to the problem of controlled convergence. Real Anal. Exch. 19 (1994), 81–97. MR 1268833
[3] R. A.  Gordon: Some comments on the McShane and Henstock integrals. Real Anal. Exch. 23 (1997), 329–341. DOI 10.2307/44152859 | MR 1609917 | Zbl 0943.26023
[4] R. A.  Gordon: The Integrals of Lebesgue, Denjoy, Perron and Henstock. Amer. Math. Soc., Providence, 1994. MR 1288751 | Zbl 0807.26004
[5] J.  Kurzweil: On multiplication of Perron integrable functions. Czechoslovak Math.  J. 23(98) (1973), 542–566. MR 0335705 | Zbl 0269.26007
[6] J.  Kurzweil, J.  Jarník: Perron type integration on n-dimensional intervals as an extension of integration of step functions by strong equiconvergence. Czechoslovak Math.  J. 46(121) (1996), 1–20. MR 1371683
[7] T. Y.  Lee: On a generalized dominated convergence theorem for the AP  integral. Real Anal. Exch. 20 (1995), 77–88. MR 1313672 | Zbl 0820.26006
[8] K.  Liao: On the descriptive definition of the Burkill approximately continuous integral. Real Anal. Exch. 18 (1993), 253–260. DOI 10.2307/44133066 | MR 1205520 | Zbl 0774.26006
[9] Y. J.  Lin: On the equivalence of four convergence theorems for the AP-integral. Real Anal. Exch. 19 (1994), 155–164. MR 1268841 | Zbl 0813.26002
[10] J. M.  Park: Bounded convergence theorem and integral operator for operator valued measures. Czechoslovak Math.  J. 47(122) (1997), 425–430. DOI 10.1023/A:1022403232211 | MR 1461422 | Zbl 0903.46040
[11] J. M.  Park: The Denjoy extension of the Riemann and McShane integrals. Czechoslovak Math.  J. 50(125) (2000), 615–625. DOI 10.1023/A:1022845929564 | MR 1777481 | Zbl 1079.28502
[12] J. M.  Park, C. G.  Park, J. B.  Kim, D. H.  Lee, and W. Y.  Lee: The $s$-Perron, sap-Perron and ap-McShane integrals. Czechoslovak Math.  J. 54(129) (2004), 545–557. DOI 10.1007/s10587-004-6407-7 | MR 2086715
[13] A. M.  Russell: Stieltjes type integrals. J.  Austr. Math. Soc. (Ser.  A) 20 (1975), 431–448. DOI 10.1017/S1446788700016153 | MR 0393379 | Zbl 0313.26012
[14] A. M.  Russell: A Banach space of functions of generalized variation. Bull. Aust. Math. Soc. 15 (1976), 431–438. DOI 10.1017/S0004972700022863 | MR 0430180 | Zbl 0333.46025
Partner of
EuDML logo