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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 dimH E = 0 onto a set F of dimH[0; 1] \ F = 0. [ABSTRACT FROM AUTHOR]

Let E be a cookie-cutter set with dim H E = 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 inf t→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 sup t→0 $$ \tfrac{{g(t)}} {{t^s }} $$ < ∞. [ABSTRACT FROM AUTHOR]

Let τ be a premeasure on a complete separable metric space and let τ* be the Method I measure constructed from τ. We give conditions on τ such that τ* has a regularity as follows: Every τ*-measurable set has measure equivalent to the supremum of premeasures of its compact subsets. Then we prove that the packing measure has this regularity if and only if the corresponding packing premeasure is locally finite. [ABSTRACT FROM AUTHOR]

Abstract.  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. [ABSTRACT FROM AUTHOR]