Trigonometric and Rademacher measures of nowhere finite variation
Author:
R. Anantharaman
Journal:
Proc. Amer. Math. Soc. 136 (2008), 31953204
MSC (2000):
Primary 46G10; Secondary 28B45
Published electronically:
May 2, 2008
MathSciNet review:
2407084
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Abstract: Let be an infinite dimensional real Banach space. It was proved by E. Thomas and soon thereafter by L. Janicka and N. J. Kalton that there always exists a measure into with relatively normcompact range such that its variation measure assumes the value on every nonnull set. Such measures have been called ``measures of nowhere finite variation'' by K. M. Garg and the author, who as well as L. Drewnowski and Z. Lipecki have done related investigations. We give some ``concrete'' examples of such 's in the spaces defined using the (real) trigonometric system and the Rademacher system illustrating similarities and some differences. We also look at the extensibility of the integration map of these 's. As an application of the trigonometric example, we have the probably known result: For every , the function is unbounded on every set with positive measure.
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 [A1]
 R. Anantharaman, Weakly null sequences in range of a vector measure and its integration map, Acta. Sci. Math. (Szeged) 70 (2004), 167182. MR 2072697 (2005d:28024)
 [A2]
 R. Anantharaman, The sequence of Rademacher averages of measurable sets, Comment. Math. Prace Mat. 30 (1990), 58 (1991). MR 1111780 (92d:46070)
 [AD]
 R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), 221235. MR 1122692 (92g:46049)
 [AG]
 R. Anantharaman and K. M. Garg, The properties of a residual set of vector measures, in Measure Theory and Its Applications, Proc. Conf. Sherbrooke, Quebec (Canada), 1982, J.M. Belly, J. Dubois and P. Morales (eds.), Lecture Notes in Math. 1033, SpringerVerlag, Berlin, 1983, 1235. MR 729522 (86d:28014)
 [D]
 J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math. 92, SpringerVerlag, New York, 1984. MR 737004 (85i:46020)
 [DU]
 J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys 15, Amer. Math. Soc., Providence, RI, 1977. MR 0453964 (56:12216)
 [DJT]
 J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge Studies in Advanced Math. 43, Cambridge Univ. Press, 1995. MR 1342297 (96i:46001)
 [DrL]
 L. Drewnowski and Z. Lipecki, On vector measures which have everywhere infinite variation or noncompact range, Dissert. Math. (Rozprawy Mat.) 339, Warsawa, 1995. MR 1316274 (96f:46085)
 [DuS]
 N. Dunford and J. Schwartz, Linear Operators. I, Interscience, New York, 1958. MR 0117523 (22:8302)
 [JK]
 L. Janicka and N. J. Kalton, Vector measures of infinite variation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 239241. MR 0444889 (56:3235)
 [KS]
 B. Kacmarz, H. Steinhaus, Theorie der Orthogonale Reihen, Chelsea reprint, 1935.
 [KK]
 I. Kluvánek and G. Knowles, Vector measures and control systems, NorthHolland, American Elsevier, AmsterdamNew York, 1976. MR 0499068 (58:17033)
 [L]
 J. Lindenstrauss, A short proof of Liapounoff's convexity theorem, J. Math. Mech. 15 (1966), 971972. MR 0207941 (34:7754)
 [LT]
 J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, SpringerVerlag, Berlin and New York, 1977. MR 0500056 (58:17766)
 [PRo]
 C. Piñerio and L. RodriguezPiazza, Banach spaces in which every compact lies inside the range of a vector measure, Proc. Amer. Math. Soc. 114 (1992)(2), 505517. MR 1086342 (92e:46038)
 [R]
 L. RodriguezPiazza, Derivability, variation and range of a vector measure, Studia Math. 112 (1995), 165187. MR 1311694 (96c:28014)
 [Ro]
 H. P. Rosenthal, On quasicomplemented subspaces of Banach spaces with an appendix on compactness of operators from to , J. Funct. Anal. 4 (1969), 176214. MR 0250036 (40:3277)
 [T]
 E. Thomas, The LebesgueNikodym theorem for vector valued measures, Mem. Amer. Math. Soc. 139 (1974).
 [Z]
 A. Zygmund, Trigonometric Series, 2nd ed., Vols. I, II, Cambridge Univ. Press, New York, 1959. MR 0107776 (21:6498)
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Additional Information
R. Anantharaman
Affiliation:
Professor Emeritus, Department of Mathematics and Computer Information Sciences, SUNY College at Old Westbury, Old Westbury, New York 115680210
Email:
rajan_a2000@yahoo.com
DOI:
http://dx.doi.org/10.1090/S0002993908092794
PII:
S 00029939(08)092794
Received by editor(s):
April 10, 2007
Received by editor(s) in revised form:
July 17, 2007
Published electronically:
May 2, 2008
Dedicated:
Dedicated to my teacher, Krishna M. Garg
Communicated by:
N. TomczakJaegermann
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
