Scaling properties of Hausdorff and packing measures

Let m is an element of N. Let theta be a continuous increasing function def ined on R+, for which theta (0) = 0 and theta (t)/t(m) is a decreasing func tion of t. Let parallel to . parallel to be a norm on R-m, and let Q, H-the ta = H-Q(theta), P-theta = P-Q(theta) denote the corresponding metric, and Hausdorff and packing measures. respectively, We characterize those functio ns theta such that the corresponding Hausdorff or packing measure scales wi th exponent alpha by showing it must be of the form theta (t) = t(alpha)L(t ), where L is slowly varying. We also show that for continuous increasing f unctions theta and eta defined on R+. for which theta (0) = eta (0) = 0. H- theta = P-eta is either trivially true or false: we show that if H-theta = P-eta. then H-theta = P-eta = c. lambda for a constant c, where lambda is t he Lebesgue measure on R-M.