A Dimension Result Arising From the Lq-Spectrum of a Measure

We give a rigorous proof of the following heuristic result: Let μ be a Borel probability measure and let ℾ(q) be the Lq-spectrum of μ . If ℾ(q) is differentiable at q = 1, then the Hausdorff dimension and the entropy dimension of μ equal ℾ1(1). Our result improves significantly some recent results of a similar nature; it is also of particular interest for computing the Hausdorff and entropy dimensions of the class of self-similar measures defined by maps which do not satisfy the open set condition.