Many important definitions in the theory of multifractal measures on ℝd, such as the Lq-spectrum, L∞-dimensions, and the Hausdorff dimension of a measure, cannot be applied to non-compactly supported or infinite measures. We propose definitions that extend the original definitions to positive Borel measures on ℝd which are finite on bounded sets, and recover many important results that hold for compactly supported finite measures. In particular, we prove that if the Lq-spectrum is differentiable at q = 1, then the derivative is equal to the Hausdorff dimension of the measure.