Hausdorff capacity and Lebesgue measure. (English) Zbl 0879.28005

Salisbury, T. S.; Steprāns, J.

Real Anal. Exch. 22(1996-97), No. 1, 265-278 (1997).

Summary: 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.

MSC:

28A12	Contents, measures, outer measures, capacities
26A03	Foundations: limits and generalizations, elementary topology of the line
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
28A35	Measures and integrals in product spaces
28A78	Hausdorff and packing measures

Keywords:

Hausdorff capacity; Lebesgue measure; continuous function