On the Π⁰_γ-completeness and Σ⁰_γ-completeness of multifractal decomposition sets

The purpose of this paper twofold. Firstly, we establish Π⁰_γ-completeness and Σ⁰_γ-completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the Π⁰_γ-completeness and Σ⁰_γ-completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if μ is a self-similar measure satisfying the strong separation condition and is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of μ defined by
{x ε ℝ^d | lim r ↘ 0 [log μ(B(x,r))/log r = α]} are Π⁰₃-complete provided they are non-empty.