It is shown that for any \(r\in (0,1)\) and for any continuous function from the unit interval to itself, there are sets of arbitrarily small Lebesgue measure whose preimage has arbitrarily large \(r\)-Hausdorff capacity. This is generalized to functions from the unit square to the interval.
"Hausdorff capacity and Lebesgue measure." Real Anal. Exchange 22 (1) 265 - 278, 1996/1997.