Title:
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On the $\sigma $-finiteness of a variational measure (English) |
Author:
|
Caponetti, Diana |
Language:
|
English |
Journal:
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Mathematica Bohemica |
ISSN:
|
0862-7959 (print) |
ISSN:
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2464-7136 (online) |
Volume:
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128 |
Issue:
|
2 |
Year:
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2003 |
Pages:
|
137-146 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
Summary:
|
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 $-finite variational measure generated by a continuous function. (English) |
Keyword:
|
variational measure |
Keyword:
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$H$-differentiable |
Keyword:
|
$H$-density |
MSC:
|
26A24 |
MSC:
|
26A39 |
MSC:
|
26A45 |
MSC:
|
28A15 |
idZBL:
|
Zbl 1027.26007 |
idMR:
|
MR1995568 |
DOI:
|
10.21136/MB.2003.134037 |
. |
Date available:
|
2009-09-24T22:07:56Z |
Last updated:
|
2020-07-29 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/134037 |
. |
Reference:
|
[1] B. Bongiorno: Essential variations.Springer Lecture Notes Math. 945 (1981), 187–193. MR 0675282 |
Reference:
|
[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. MR 1407278, 10.2307/44152676 |
Reference:
|
[3] B. Bongiorno, L. Di Piazza, V. Skvortsov: On variational measures related to some bases.J. Math. Anal. Appl. 250 (2000), 533–547. MR 1786079, 10.1006/jmaa.2000.6983 |
Reference:
|
[4] B. Bongiorno, L. Di Piazza, D. Preiss: Infinite variation and derivatives in $\mathbb{R}^m$.J. Math. Anal. Appl. 224 (1998), 22–33. MR 1632942, 10.1006/jmaa.1998.5982 |
Reference:
|
[5] Z. Buczolich, W. F. Pfeffer: When absolutely continuous implies $\sigma $-finite.Bull. Csi., Acad. Royale Belgique, serie 6 (1997), 155–160. MR 1625113 |
Reference:
|
[6] Z. Buczolich, W. F. Pfeffer: Variations of additive functions.Czechoslovak Math. J. 47 (1997), 525–555. MR 1461431, 10.1023/A:1022471719916 |
Reference:
|
[7] J. L. Doob: Measure Theory.Springer, New-York, 1994. Zbl 0791.28001, MR 1253752 |
Reference:
|
[8] L. Di Piazza: Variational measures in the theory of the integration in $\mathbb{R}^m$.Czechoslovak Math. J. 51 (2001), 95–110. MR 1814635, 10.1023/A:1013705821657 |
Reference:
|
[9] V. Ene: Thomson’s variational measure and nonabsolutely convergent integrals.Real Anal. Exch. 26 (2000/01), 35–50. MR 1825496 |
Reference:
|
[10] C.-A. Faure: A descriptive definition of the KH-Stieltjes integral.Real Anal. Exch. 23 (1997/98), 113–124. MR 1609775 |
Reference:
|
[11] P. Mattila: Geometry of sets and measures in Euclidean spaces.Cambridge University Press, 1995. Zbl 0819.28004, MR 1333890 |
Reference:
|
[12] W. F. Pfeffer: The Riemann Approach to Integration.Cambridge University Press, 1993. Zbl 0804.26005, MR 1268404 |
Reference:
|
[13] W. F. Pfeffer: On additive continuous functions of figures.Rend. Istit. Mat. Univ. Trieste, suppl. (1998), 115–133. Zbl 0921.26008, MR 1696024 |
Reference:
|
[14] W. F. Pfeffer: The Lebesgue and Denjoy-Perron integrals from a descriptive point of view.Ricerche Mat. 48 (1999), 211–223. Zbl 0951.26005, MR 1760817 |
Reference:
|
[15] C. A. Rogers: Hausdorff Measures.Cambridge, 1970. Zbl 0204.37601, MR 0281862 |
Reference:
|
[16] S. Saks: Theory of the Integral.Dover, New York, 1964. MR 0167578 |
Reference:
|
[17] B. S. Thomson: Derivates of interval functions.Mem. Amer. Math. Soc., Providence 452 (1991). Zbl 0734.26003, MR 1078198 |
Reference:
|
[18] B. S. Thomson: Some properties of variational measures.Real Anal. Exch. 24 (1998/99), 845–853. MR 1704758, 10.2307/44153004 |
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