Hausdorff measures of different dimensions are not Borel isomorphic

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (R, B, H) and (R, B, H) are not isomorphic if s 6= t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of R and H is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for every Borel function f : R → R and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that f(C) has Hausdorff dimension ≤ d. We also prove this statement in a more general form: If A ⊂ R is Borel and f : A→ R is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dimB = d ·dimA and dim f(B) ≤ d · dim f(A).