Trigonometric and Rademacher measures of nowhere finite variation

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.