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We prove that there is a homeomorphism of the unit interval onto itself that is so singular that it maps some set $E$ of $\dim_HE=0$ onto a set $F$ of $\dim_H[0,1]\setminus F=0$.

For every doubling gauge g, we prove that there is a Cantor set of positive finite -measure, -measure, and -premeasure. Also, we show that every compact metric space of infinite -premeasure has a compact countable subset of infinite -premeasure. In addition, we obtain a class of uniform Cantor sets and prove that, for every set E in this class, there exists a countable set F, with , and a doubling gauge g such that has different positive finite -measure and -premeasure.

Let X be an Ahlfors d-regular space and m a d-regular measure on X. We prove that a measure µ on X is d-homogeneous if and only if µ is mutually absolutely continuous with respect to m and the derivative Dmµ(x) is an A1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.

Let E be a cookie-cutter set with dimHE = s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 < lim inft→0\( \tfrac{{g(t)}} {{t^s }} \) < ∞. Also, we prove that for a doubling gauge function g the set E has positive and finite packing g-measure if and only if 0 < lim supt→0\( \tfrac{{g(t)}} {{t^s }} \)< ∞.

We construct a sequence of measurable functions converging at each point of the unit interval, but the set of points with any given rate of convergence has Hausdorff dimension one. This is used to show that a version of Egoroff’s theorem due to Taylor is best possible. The construction relies on an analysis of the maximal run length of ones in the dyadic expansion of real numbers. It is also proved that the exceptional set for a limit theorem of Renyi has Hausdorff dimension one.

Let g be a doubling gauge. We consider the packing measureP g and the packing premeasureP g in a metric space X. We rst show that ifP g 0 (X) is nite, then as a function of X,P g has a kind of \outer regularity". Then we prove that if X is complete separable, then supP g 0 (F ) P g (B) supP g 0 (F ) for every Borel subset B of X, where the supremum is taken over all compact subsets of B having nite P g -premeasure, and is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling gauge function, there is a compact metric space of nite positive packing measure.