Trigonometric and Rademacher measures of nowhere finite variation

Author:
R. Anantharaman

Journal:
Proc. Amer. Math. Soc. **136** (2008), 3195-3204

MSC (2000):
Primary 46G10; Secondary 28B45

DOI:
https://doi.org/10.1090/S0002-9939-08-09279-4

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 norm-compact range such that its variation measure assumes the value on every non-null 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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Additional Information

**R. Anantharaman**

Affiliation:
Professor Emeritus, Department of Mathematics and Computer Information Sciences, SUNY College at Old Westbury, Old Westbury, New York 11568-0210

Email:
rajan_a2000@yahoo.com

DOI:
https://doi.org/10.1090/S0002-9939-08-09279-4

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. Tomczak-Jaegermann

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.