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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
Published electronically: May 2, 2008
MathSciNet review: 2407084
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Abstract: Let $ X$ 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 $ \mu$ into $ X$ with relatively norm-compact range such that its variation measure assumes the value $ \infty$ 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 $ \mu$'s in the $ l^p$ spaces defined using the (real) trigonometric system $ (t_n)$ and the Rademacher system $ (r_n)$ illustrating similarities and some differences. We also look at the extensibility of the integration map of these $ \mu$'s. As an application of the trigonometric example, we have the probably known result: For every $ p\ge1$, the function $ (\Sigma (\vert t_n (t) \vert ^p ) / n )$ is unbounded on every set $ A$ 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

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.