Trigonometric and Rademacher measures of nowhere finite variation
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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 $p\ge 1$, the function $(\Sigma (| t_n (t) | ^p ) / n )$ is unbounded on every set $A$ with positive measure.References
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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
- Received by editor(s): April 10, 2007
- Received by editor(s) in revised form: July 17, 2007
- Published electronically: May 2, 2008
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2407084
Dedicated: Dedicated to my teacher, Krishna M. Garg