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Let X be an Ahlfors d-regular space and m a d-regular measure on X. We prove that a measure µ on X is d-homogeneous if and only if µ is mutually absolutely continuous with respect to m and the derivative Dmµ(x) is an A1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.

Let E be a cookie-cutter set with dimHE = s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 < lim inft→0\( \tfrac{{g(t)}} {{t^s }} \) < ∞. Also, we prove that for a doubling gauge function g the set E has positive and finite packing g-measure if and only if 0 < lim supt→0\( \tfrac{{g(t)}} {{t^s }} \)< ∞.