摘要 : We show that the Continuum Hypothesis implies that for every 0 < d_1 ≤ d_2 ≤ n the measure spaces (R~n, M_(H~(d_1)),H~(d_1)) and (R~n,M_(H~(d_2)) ,H~(d_2)) are isomorphic, where H~d is d-dimensional Hausdorff measure and M_(H~d) i... 展开 We show that the Continuum Hypothesis implies that for every 0 < d_1 ≤ d_2 ≤ n the measure spaces (R~n, M_(H~(d_1)),H~(d_1)) and (R~n,M_(H~(d_2)) ,H~(d_2)) are isomorphic, where H~d is d-dimensional Hausdorff measure and M_(H~d) is the σ-algebra of measurable sets with respect to H~d. This is motivated by the well-known question (circulated by D. Preiss) whether such an isomorphism exists if we replace measurable sets by Borel sets. We also investigate the related question whether every continuous function (or the typical continuous function) is Hoelder continuous (or is of bounded variation) on a set of positive Hausdorff dimension. 收起