Trigonometric and Rademacher measures of nowhere finite variation

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.