[1] introduced fractal geometric entropies and dimensions for Voiculescu’s microstate spaces ([3], [4]). One can associate to a finite set of selfadjoint elements X in a tracial von Neumann algebra and an α > 0 an extended real number H(X) ∈ [−∞,∞]. H(X) is a kind of asymptotic logarithmic α-Hausdorff measure of the microstate spaces of X. One can also define a free Hausdorff dimension of X, de...

Critical circle homeomorphisms have an invariant measure totally singular with respect to the Lebesgue measure. We prove that singularities of the invariant measure are of Hőlder type. The Hausdorff dimension of the invariant measure is less than 1 but greater than 0.

The uniformity invariant for Lebesgue measure is defined to be the least cardinal of a non-measurable set of reals, or, equivalently, the least cardinal of a set of reals which is not Lebesgue null. This has been studied intensively for the past 30 years and much of what is known can be found in [?] and other standard sources. Among the well known results about this cardinal invariant of the co...

How do we measure the ”size” of a set in IR? Let’s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its length, which is used frequently in differentiation and integration. For any bounded interval I (open, closed, half-open) with endpoints a and b (a ≤ b), the length of I is defined by `(I) = b − a. Of course, the length of any unbounded...

In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman’s lemma and show that Hausdorff dimension can be computed through a counting argument rather than by taking the infimum of a sum of certain covers. This formulation is then applied to obtain a ...

(i= 1 . 2, . . .) such that the intersection n A, contains a perfect subset i=1 (and is therefore of power 2No) . They asked for what Hausdorff measure functions (k(i) is it possible to choose the subsequence to make the intersection set (1 A„,, of positive -measure . In the present note We show that the strongest possible result in this direction is true . This is given by the following; theor...

Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals which are random for an arbitrary (continuous) probability measure or a generalized Hausdorff measur...

This paper is mainly concerned with scrambled sets for Luroth map. It shown that all have null Lebesgue measure and there exists a set full Hausdorff dimension.