On s-sets and mutual absolute continuity of measures on homogeneous spaces.

We extend a recent result of A. Jonsson about mutual absolute continuity of two D-s-measures on an s-set F subset of R-n to the homogeneous spaces (X, d, mu) of Coifman, Weiss. Here we define Hausdorff measure, Hausdorff dimension, D-s-set and d-set relative to the measure mu. Our main result holds for so called (s, d)-sets, d greater than or equal to s, and is stronger than Jonssons result even in R-n. As applications we interpret this Hausdorff dimension as a relative dimension for very regular sets and show that it in general depends strongly on mu. For this purpose we construct a strictly increasing function f : R --> R, whose measure is doubling and concentrated on a set of arbitrary small Hausdorff dimension. The extension of f to a quasiconformal map of the half plane onto itself sharpens a classical example of Ahlfors-Beurling.