We study the typical behaviour (in the sense of Baire's category) of the multifractal box dimensions of measures on $ℝ^{d}$. We prove that in many cases a typical measure μ is as irregular as possible, i.e. the lower multifractal box dimensions of μ attain the smallest possible value and the upper multifractal box dimensions of μ attain the largest possible value.
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For a subset $E ⊆ ℝ^{d}$ and $x ∈ ℝ^{d}$, the local Hausdorff dimension function of E at x is defined by $dim_{H,loc}(x,E) = lim_{r↘ 0} dim_{H}(E ∩ B(x,r))$ where $dim_{H}$ denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.
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