A dimension result arising from the $L^{q}$-spectrum of a measure

We give a rigorous proof of the following heuristic result: Let $\mu$ be a Borel probability measure and let $\tau (q)$ be the $L^{q}$-spectrum of $\mu$. If $\tau (q)$ is differentiable at $q=1$, then the Hausdorff dimension and the entropy dimension of $\mu$ equal $\tau '(1)$. 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.