Let ( be a Borel Probability measure on R^d. q, t,∈ R. Let H_(^(q,t) denote the multifractal Hausdorff measure. We prove that, when satisfies the so-called Federer condition, for a closed subset E∈R^n, such that H_(^(q,t) (E) > 0, there exists a compact subset F of E with 0 < H_(^(q,t) (F) <∞ , i.e, the finite measure subsets of multifractal Hausdorff measure exist.
