Bassed on the intrinsic structure of a selfmapping T: S → S of an arbitrary set S, called the orbit-structure of T, a new entropy is defined. The idea is to use the number of preimages of an element x under the iterates of T to characterize the complexity of the transformation T and their orbit graphs. The fundamental properties of the orbit entropy related to iteration, iterative roots and iteration semigroups are studied. For continuous (differentiable) functions of Rn to Rn, the chaos of Li and Yorke is characterized by means of this entropy, mainly using the method of Straffingraphs.

If E is a subset of ℝn (n ≥ 1) we define the distance set of E as

The best known result on distance sets is due to Steinhaus [11], namely, that, if E ⊂ ℝn is measurable with positive n-dimensional Lebesgue measure, then D(E) contains an interval [0, ε) for some ε > 0. A number of variations of this have been examined, see Falconer [6, p. 108] and the references cited therein.

It is shown that a weakly compact convex set in a locally convex space is a zonoform if and only if it is the order continuous image of an order interval in a Dedekind complete Riesz space. While this result implies the Kluv´nek characterization of the range of a vector measure, the techniques of the present paper are purely order theoretic.

If E1 and E2 are subsets of ℝn and a- is an isometry or similarity transformation, it is useful to be able to estimate the Hausdorff dimension of E1 ∩ σ(E2) in terms of the dimensions of E1 and E2. If E1 and E2 are compact, then, as σvaries, dim (El ∩ σ(E2)) is “in general” at most max (dim E1 + dim E2 − n, 0) and “often” at least this value (see Mattila [9] and Kahane [7] for more precise statements of these ideas). However, as we shall see, it is possible to construct non-compact sets E of any given dimension that are “sufficiently dense” in ℝn to ensure that dim (E ∩ σ(E)) = dim E for all similarities σ More generally, we shall show that for each s there are large classes of sets & of dimensions between s and n, closed under reasonable transformations including similarities, such that the intersection of any countable collection of sets in & has dimension at least s. Such collections of sets are required, for example, in the constructions of subsets of ℝn with certain dimensional properties given by Davies [1] and Falconer [5].

Si E et F sont deux espaces vectoriels en dualité séparante, M+(E, F) désigne le cône des mesures coniques positives sur E mis en dualité avec F, c'est à dire le cônes des formes postives sur le treillis de fonctions sur E engendré par F. Ce sont des objets plus généraux que les mesures cylindriques admettant des moments finis d'ordre un.

On part d'abord d'une mesure conique représentée par une mesure de Radon sur le complété faible de E et on donne des critéres (par exemple R.N.P.) pour qu'elle le soit sur l'espace E lui-même.

On étudie ensuite les cônes faiblement complets saillants (classe L) contenus dans un espace de Banach ou dans le dual d'un espace de Fréchet F; on montre notamment qu' un cône faiblement fermé contenu dans F′ est dans Lsi son polaire dans F est positivement engendré.

Si B est un espace de Banach et 11 ⊄ B, on cherche à prologner une μ ∈ M+(B′, B) en un élement de M+ (B′, B″). On montre également que, si X est un convexe compact, toute fonction vérifiant le calcul barycentrique sur X est continue sur des ensembles fixes que l'on précise.

Enfin on donne des conditions (de type bornologique) sur un e.l.c.s E, permettant d'interpréter une μ ∈ M+ (E, E′) comme une mesure conique sur un espace normé.

A basic notion in the classical theory of differentiation is that of a differentiation base. However, some differentiation type theorems only require the less restricted notion of a contraction. We demonstrate that the classical criteria, such as the covering criteria of de Possel, continue to hold in the new setting.

The results we present were motivated by the product measure problem for Baire measures. For two completely regular Hausdorff spaces X and Y, with totally finite a- additive measures μ and ν defined on the Baire σ- algebras ℬ0(X) and ℬ0(Y) respectively, under what conditions may we define a measure λ on the Baire σ-algebra ℬ0(X × Y), extending the product measure μ ⊗ ν defined on the product σ-algebra ℬ0(X) × ℬ0(Y) and satisfying a Fubini theorem?

Cardinal functions of topologies have been extensively studied. Cardinal functions of measures have attracted less interest, perhaps because there are fewer straightforward results which are independent of special axioms. In this paper I consider the “additivity” and “cofinality” of a measure (Definition 1) and show that they can often be calculated in terms of certain fundamental cardinals (Corollary 11 and Theorem 16).

V. Krishnamurthy has shown that on a finite set X all topologies can be mapped into a certain set of matrices of zeros and ones. In this paper it is shown that all lattices, algebras and rings on a finite set X can be mapped onto particular sets of matrices of zeros and ones.

In 1942 Piccard [10] gave an example of a set of real numbers whose sum set has zero Lebesgue measure but whose difference set contains an interval. About thirty years later various authors (Connolly, Jackson, Williamson and Woodall) in a series of papers constructed F σ sets E in ℝ such that E – E contains an interval while the K-fold sum set

has zero Lebesgue measure for progressively larger values of k.

It is shown that a positive measure μ on the Borel subsets of Rk is translation-bounded if and only if the Fourier transform of the indicator function of every bounded Borel subset of Rk belongs to L2(μ).

We define s-dimensional Hausdorff outer measure on subsets of ℝn by

Here | | denotes the diameter of a set. The Souslin sets, which include the Borel sets, are Hs-measurable for any s ≥ 0.

In July 1982, I was asked by Prof. Jorgen Hoffmann-Jorgensen to construct an uncountable compact set K in the line which was symmetric about 0 and had the property that, for all n, the set of sums of n-tuples from K has measure 0. There are two equivalent conditions: the set of such sums should never contain an interval, or K* ≠ ℝ, where K* is the subgroup of (ℝ, +) generated by K. I did so, and the set I constructed had entropy dimension 0 (and thus also Hausdorff dimension 0). Hoffmann-Jorgensen showed that every set of entropy dimension 0 would exhibit the same behaviour. However, I did not believe that the essence of the example lay in its dimension, and I here modify my construction so that the set K has dimension 1 (and thus also entropy dimension 1), while K* ≠ ℝ, as before. By contrast, the Cantor ternary set has dimension log3(2), but the set of differences is the interval [ –1, 1], so that it does generate ℝ. It follows that the property under consideration is arithmetical rather than dimensional.

Let ℒ be a lattice of subsets of a set X. Let MR (ℒ) denote the set of all ℒ-regular (finitely additive) measures on the algebra generated by ℒ. Under the assumption that ℒ is disjunctive, in the first part of the paper, criteria are obtained for the σ-smoothness, τ-smoothness, and tightness of elements of MR(ℒ) in terms of the general Wallman remander. in the second part of the paper, various applications are given, and, in particular, extensions and refinements of the Yoside-Hewitt Decom position Theorem are obtained.

Let T: [0,1] → [0,1] be a map which is given piecewise as a linear fractional map such that T0 = T] = 0 and T'0 < 1. Then T is ergodic and admits an invariant measure which can be calculated explicitly.

It is shown that hte algebra of regular sets for a finitely additive Borel measure μ on a compact Hausdroff space is a σ-algebra only if it includes the Baire algebra and μ is countably additive onthe σ-algebra of regular sets. Any infinite compact Hausdroff space admits a finitely additive Borel measure whose algebra of regular sets is not a σ-algebra. Although a finitely additive measure with a σ-algebra of regular sets is countably additive on the Baire σ-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a σ-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, 1}-valued countably additive measure from a σ-algebra to a larger σ-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex.

For a metric space <Ω, ρ> and a ‘measure function’ h, the Hausdorff measure mh on Ω is denned by applying Method II to the premeasure defined by τ(E) = h(d(E)), E ⊆ Ω, where

with d(Φ) = 0, is the diameter of E. The set function mh is then a metric outer measure. There are many variations on this definition producing measures also associated with the name Hausdorff. Here we are concerned with those measures which arise when there is a restriction on the sets E for which τ is defined. Such measures arise, for example, as net measures, Rogers [1]. Also we might find it useful to have τ defined only on disks, or only on squares, or only on rectangles with a given relation between vertical and horizontal sides.

Let u be a solution of the heat equation which can be written as the difference of two non-negative solutions, and let v be a non-negative solution. A study is made of the behaviour of u(x, t)/v(x, t) as t → 0+. The methods are based on the Gauss-Weierstrass integral representation of solutions on Rn × ]0, a[ and results on the relative differentiation of measures, which are employed in a novel way to obtain several domination, non-negativity, uniqueness and representation theorems.

Given a simplex S and a positive function δ on S, we show that there is a simplicial subdivision of S such that the diameter of each subdividing simplex is smaller that δ evaluated at some of its vertices.

This paper is concerned with two aspects of the theory of measures on compact totally ordered spaces (the topology is to be the order topology). In Section 2, we clarify a recent construction of Sapounakis [11, 12] and, in so doing, we are able to say a little more about it. It should be added here that Sapounakis had other ends in view. To be precise, let I be the closed unit interval [0, 1] and let λ be Lebesgue measure on I. We shall construct another totally ordered set Ĩ which is compact in its order topology, a continuous increasing surjection τ : Ĩ → I with the property that card τ−1(t) = 2 for all t ∈ ]0,1[ (these brackets denote the open interval), and a measure on Ĩ such that τ() = λ. Then the following theorem holds.