Finitely subadditive and countably subadditive outer measures

Results are given comparing countably subadditive (csa) outer measures and finitely subadditive (fsa) outer measures, especially relating to regularity and measurability conditions such as (*) condition:A setE (of an arbitrary setX), is measurable ( an outer measure),ES (the collection of measurable sets) iff (X)=(E)+(E). Specific examples are given contrasting csa and fsa outer measures. In particular fsa and csa outer measures derived from finitely additive measures defined on an algebra of sets generated by a lattice of sets, are investigated in some detail.     

Tangent measure distributions of fractal measures

Tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in Euclidean spaces. The idea is, loosely speaking, to attach to every point of the set a family of random measures, called the -dimensional tangent measure distributions at the point, which describe asymptotically the -dimensional scenery seen by an observer zooming down towards this point. This tool has been used by Bandt [BA] and Graf [G] to study the regularity of the local geometry of self similar sets, but in this paper we show that its scope goes much beyond this situation and, in fact, it may be used to describe a strong regularity property possessed by every measure: We show that, for every measure on a Euclidean space and any dimension , at -almost every point, all -dimensional tangent measure distributions are Palm measures. This means that the local geometry of every dimension of general measures can be described – like the local geometry of self similar sets – by means of a family of statistically self similar random measures. We believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities. Received: 27 November 1996 / Revised version: 27 February 1998     

Conical upper density theorems and porosity of measures

We study how measures with finite lower density are distributed around (n−m)-planes in small balls in Rn. We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.     

On characterization and properties of replete Wallman spaces

The subset of lattice regular zero-one valued measures on an algebra generated by a lattice (a Wallman-type space) which integrates all lattice continuous functions on an arbitrary setX is introduced and some properties of it are presented.Then repleteness of certain Wallman spaces is considered and used in finally establishing conditions whereby the space of lattice regular zero-one valued measures on the algebra generated by a lattice which are countably additive (a Wallman-type space) is realcompact.     

Global properties of invariant measures

We study global regularity properties of invariant measures associated with second order differential operators in R^N. Under suitable conditions, we prove global boundedness of the density, Sobolev regularity, a Harnack inequality and pointwise upper and lower bounds.     

The doubling property of conformal measures of infinite iterated function systems

We provide sufficient conditions for the conformal measures induced by regular conformal infinite iterated function systems to satisfy the doubling property. We apply these conditions to iterated function systems derived from the continued fraction algorithm—continued fractions with restricted entries. For these systems our conditions are expressed in terms of the asymptotic density properties of the allowed entries. As examples, we give some relatively large classes of sets of continued fractions with restricted entries for which the corresponding conformal measures have the doubling property. Similarly, we give some other classes for which the conformal measure does not have the doubling property.     

Measure theory based on lattices and transfinite recursion

In this methodological study we develop the foundations of measure theory using lattices as prime structures instead of rings. Topological as well as abstract regularity is incorporated into this approach from the outset. The use of inner and outer measures is replaced by transfinite constructions. Basic extension steps are transfinitely iterated to yield generalizations of Carathéodory’s theorem which are optimal with respect to inner and outer approximations. Received: 5 May 2008     

Scaling properties of Hausdorff and packing measures

Let . Let be a continuous increasing function defined on , for which and is a decreasing function of t. Let be a norm on , and let , , denote the corresponding metric, and Hausdorff and packing measures, respectively. We characterize those functions such that the corresponding Hausdorff or packing measure scales with exponent by showing it must be of the form , where L is slowly varying. We also show that for continuous increasing functions and defined on , for which , is either trivially true or false: we show that if , then for a constant c, where is the Lebesgue measure on . Received June 17, 2000 / Accepted September 6, 2000 / Published online March 12, 2001     

Vector measures and strict topologies

Let X be a completely regular Hausdorff space and Cb(X) be the space of all real-valued bounded continuous functions on X, endowed with the strict topology βσ. We study topological properties of continuous and weakly compact operators from Cb(X) to a locally convex Hausdorff space in terms of their representing vector measures. In particular, Alexandrov representation type theorems are derived. Moreover, a Yosida-Hewitt type decomposition for weakly compact operators on Cb(X) is given.     

Some remarks on the existence of doubly stochastic measures with latticework hairpin support

Summary In their paper Properties of a special class of doubly stochastic measures, which appeared in volume 36 of this journal (pp. 212–229), A. Kaminski et al. introduced latticework hairpins and used them to provide a counterexample to a conjecture of J. H. B. Kemperman. However, their paper contains an error and, as a consequence, a umber of incorrect statements (which fortunately do not invalidate its main results). We point out the error and the incorrect statements. More importantly, we modify the argument, present correct versions wherever possible, and provide a valid characterization of those latticework hairpins that support doubly stochastic measures. We also construct a number of new examples, including another counterexample to Kemperman's conjecture.     

Local dimensions of sliced measures and stability of packing dimensions of sections of sets

Let m and n be integers with 0<m<n. We relate the absolutely continuous and singular parts of a measure μ on to certain properties of plane sections of μ. This leads us to prove, among other things, that the lower local dimension of (n−m)-plane sections of μ is typically constant provided that the Hausdorff dimension of μ is greater than m. The analogous result holds for the upper local dimension if μ has finite t-energy for some t>m. We also give a sufficient condition for stability of packing dimensions of section of sets.     

Hausdorff measures for a class of homogeneous Moran sets

This paper provides an explicit formula for the Hausdorff measures of a class of regular homogeneous Moran sets. In particular, this provides, for the first time, an example of an explicit formula for the Hausdorff measure of a fractal set whose Hausdorff dimension is greater than 1.     

On multifractality and time subordination for continuous functions

Let Z:[0,1]→R be a continuous function. This paper relates to the existence of a decomposition of Z as Z=g○f, where g:[0,1]→R is a monofractal function with exponent 0<H<1 and f:[0,1]→[0,1] is a time subordinator, i.e. the integral of a positive Borel measure supported by [0,1]. An equivalent question consists of searching for a (multifractal) parametrization of Z which transforms Z into a monofractal function. We establish that such a decomposition can be found for a large class of functions which includes the usual examples of multifractal functions.We find an interesting relationship between self-similar functions and self-similar measures as an application of our results.Our theorems yield new insights in the understanding of the multifractal behaviour of functions, giving a significant role to the regularity analysis of Borel measures.     

Dynamics of hybrid systems induced by operator valued measures

In this paper we consider a class of hybrid systems induced by operator valued measures. This includes semigroups of operators perturbed by bounded as well as unbounded operator valued measures. We construct an evolution operator for the hybrid system and based on its properties we prove existence, uniqueness and regularity properties of solutions. We also consider Semilinear Problems driven by vector measures. Nonstandard problems arising in the study of the classical linear quadratic regulator problem in the present setting are discussed and partial solutions provided.     

On the Radon-Nikodym theorem for measures with values in vector lattices

Measures with values in a countably order-complete vector lattice are considered. The underlying σ-algebra is assumed to be σ-isomorphic to the Borel sets of the real line. Given one such measure, densities are searched which are not necessarily scalar-valued for smaller measures. The results can be used to prove the existence of a least upper bound for two such measures.     

Outer Minkowski content for some classes of closed sets

We find conditions ensuring the existence of the outer Minkowski content for d-dimensional closed sets in , in connection with regularity properties of their boundaries. Moreover, we provide a class of sets (including all sufficiently regular sets) stable under finite unions for which the outer Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positive reach belong to this class.     

Some remarks on countably determined measures and uniform distribution of sequences

Two topics are investigated: countably determined (regular Borel probability) measures on compact Hausdorff spaces, and uniform distribution of sequences regarding mainly this kind of measures. We prove several characterizations of countably determined measures, and apply the results in order to show the existence of a well distributed sequence in the support of a countably determined measure. We also generalize a result of Losert on the existence of uniformly distributed sequences in compact dyadic spaces.     

Projective limits of complex measures and martingale convergence

In this paper we prove, for signed or complex Radon measures on completely regular spaces, the analogue of Prokhorov's criterion on the existence of the projective limit of a compatible system of measures. Because of loss of mass under projections this cannot be reduced to the case of positive measures. Countable projective limits are, as in the case of positive measures, particularly simple, the sole condition now being the boundedness of the total variations. It is shown, with the help of the martingale convergence theorem, that the densities of these complex measures with respect to their variations, converge in an appropriate sense. This work is part of an extended project on the mathematical theory of path integrals. Received: 18 June 1997 / Revised version: 11 August 2000/?Published online: 9 March 2001