Multifractal Structure of Noncompactly Supported Measures

Many important definitions in the theory of multifractal measures on ^d, such as the L^q-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 ^d 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 L ^q-spectrum is differentiable at q = 1, then the derivative is equal to the Hausdorff dimension of the measure.