Cardinal Invariants Associated with Hausdorff Capacities

Abstract

This is a revision (and partial retraction) of my previous abstarct. Let $\lambda(X)$ denote Lebesgue measure. If $X\subseteq [0,1]$ and $r \in (0,1)$ then the $r$-Hausdorff capacity of $X$ is denoted by $H^r(X)$ and is defined to be the infimum of all $\sum_{i=0}^\infty \lambda(I_i)^r$ where $\{I_i\}_{i\in\omega}$ is a cover of $X$ by intervals. The $r$ Hausdorff capacity has the same null sets as the $r$-Hausdorff measure which is familiar from the theory of fractal dimension. It is shown that, given $r < 1$, it is possible to enlarge a model of set theory, $V$, by a generic extension $V[G]$ so that the reals of $V$ have Lebesgue measure zero but still have positive $r$-Hausdorff capacity.