On the $\sigma $-finiteness of a variational measure
Mathematica Bohemica, Tome 128 (2003) no. 2, pp. 137-146

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MR Zbl
The $\sigma $-finiteness of a variational measure, generated by a real valued function, is proved whenever it is $\sigma $-finite on all Borel sets that are negligible with respect to a $\sigma $-finite variational measure generated by a continuous function.
The $\sigma $-finiteness of a variational measure, generated by a real valued function, is proved whenever it is $\sigma $-finite on all Borel sets that are negligible with respect to a $\sigma $-finite variational measure generated by a continuous function.
DOI : 10.21136/MB.2003.134037
Classification : 26A24, 26A39, 26A45, 28A15
Keywords: variational measure; $H$-differentiable; $H$-density
Caponetti, Diana. On the $\sigma $-finiteness of a variational measure. Mathematica Bohemica, Tome 128 (2003) no. 2, pp. 137-146. doi: 10.21136/MB.2003.134037
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[1] B. Bongiorno: Essential variations. Springer Lecture Notes Math. 945 (1981), 187–193. | MR

[2] B. Bongiorno, L. Di Piazza, V. Skvortsov: A new full descriptive characterization of Denjoy-Perron integral. Real Anal. Exch. 21 (1995/96), 656–663. | DOI | MR

[3] B. Bongiorno, L. Di Piazza, V. Skvortsov: On variational measures related to some bases. J. Math. Anal. Appl. 250 (2000), 533–547. | DOI | MR

[4] B. Bongiorno, L. Di Piazza, D. Preiss: Infinite variation and derivatives in $\mathbb{R}^m$. J. Math. Anal. Appl. 224 (1998), 22–33. | DOI | MR

[5] Z. Buczolich, W. F. Pfeffer: When absolutely continuous implies $\sigma $-finite. Bull. Csi., Acad. Royale Belgique, serie 6 (1997), 155–160. | MR

[6] Z. Buczolich, W. F. Pfeffer: Variations of additive functions. Czechoslovak Math. J. 47 (1997), 525–555. | DOI | MR

[7] J. L. Doob: Measure Theory. Springer, New-York, 1994. | MR | Zbl

[8] L. Di Piazza: Variational measures in the theory of the integration in $\mathbb{R}^m$. Czechoslovak Math. J. 51 (2001), 95–110. | DOI | MR

[9] V. Ene: Thomson’s variational measure and nonabsolutely convergent integrals. Real Anal. Exch. 26 (2000/01), 35–50. | MR

[10] C.-A. Faure: A descriptive definition of the KH-Stieltjes integral. Real Anal. Exch. 23 (1997/98), 113–124. | MR

[11] P. Mattila: Geometry of sets and measures in Euclidean spaces. Cambridge University Press, 1995. | MR | Zbl

[12] W. F. Pfeffer: The Riemann Approach to Integration. Cambridge University Press, 1993. | MR | Zbl

[13] W. F. Pfeffer: On additive continuous functions of figures. Rend. Istit. Mat. Univ. Trieste, suppl. (1998), 115–133. | MR | Zbl

[14] W. F. Pfeffer: The Lebesgue and Denjoy-Perron integrals from a descriptive point of view. Ricerche Mat. 48 (1999), 211–223. | MR | Zbl

[15] C. A. Rogers: Hausdorff Measures. Cambridge, 1970. | MR | Zbl

[16] S. Saks: Theory of the Integral. Dover, New York, 1964. | MR

[17] B. S. Thomson: Derivates of interval functions. Mem. Amer. Math. Soc., Providence 452 (1991). | MR | Zbl

[18] B. S. Thomson: Some properties of variational measures. Real Anal. Exch. 24 (1998/99), 845–853. | DOI | MR

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