Gauges for the cookie-cutter sets

Let E be a cookie-cutter set with dim _H E = 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 inf _t→0 $ \tfrac{{g(t)}} {{t^s }} $ \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 sup _t→0 $ \tfrac{{g(t)}} {{t^s }} $ \tfrac{{g(t)}} {{t^s }} < ∞.