Some remarks on absolute continuity and quantization of probability measures

Abstract

We introduce a notion of monotonicity of dimensions of measures. We show that the upper and lower quantization dimensions are not monotone. We give sufficient conditions in terms of so-called vanishing rates such that ν ≪ μ implies D ¯ r ( ν ) ⩽ D ¯ r ( μ ) . As an application, we determine the quantization dimension of a class of measures which are absolutely continuous w.r.t. some self-similar measure, with the corresponding Radon–Nikodym derivative bounded or unbounded. We study the set of quantization dimensions of measures which are absolutely continuous w.r.t. a given probability measure μ. We prove that the infimum on this set coincides with the lower packing dimension of μ. Furthermore, this infimum can be attained provided that the upper and lower packing dimensions of μ are equal.