§1. Introduction. Let two probability spaces (X, , μ,) and (Y, ℬ, ν) be given. For a subset D of X × Y and a real number d ≥ 0 we consider the following problem

(MP) Does there exist a measure » on X × Y having μ and ν as marginals and such that λ (D) ≥ 1 − d?

This problem comes from Strassen's paper [12], where Borel probabilities on Polish spaces were treated. Further, it was investigated by many authors in more general settings (cf. [2], [4]-[7], [11]-[13]).

Various continuity conditions (in norm, in measure, weakly etc.) for the nonlinear superposition operator F x(s) = f(s, x(s)) between spaces of measurable functions are established in terms of the generating function f = f(s, u). In particular, a simple proof is given for the fact that, if F is continuous in measure, then f may be replaced by a function f which generates the same superposition operator F and satisfies the Carathéodory conditions. Moreover, it is shown that integral functional associated with the function f are proved.

The Hausdorff dimension has been used for many years for assessing the sizes of sets in Euclidean and other metric spaces, see, for example, [1,2,5,6,8,10]. However, different sets with the same Hausdorff dimension may have very different characteristics, for example, a straight line segment in ℝ2 and the Cartesian product in ℝ2 of two suitably chosen Cantor sets in ℝ will both have Hausdorff dimension 1. In this paper we develop a measure-theoretic method of distinguishing between the sets of such pairs.

Let μ, be a positive Radon measure with compact support in the euclidean n-space ℝn. Introducing the Fourier transform

and the averages over the spheres

we can write the α-energy, 0 < α < n, of μ as

where the positive constants c1 and c2 depend only on n and α. The second equality is based on the Plancherel formula and the fact that where .

We construct a universal function φ on the real line such that, for every continuously differentiable function f the range of f – φ has measure zero. We then apply this to obtain results on curve packing that generalize the Besicovitch set. In particular, we show that given a continuously differentiable family of measurable curves, there exists a plane set of measure zero containing a translate of each curve in the family. Examples are given to show that the differentiability hypothesis cannot be weakened to a Lipschitz condition of order α for any 0<α<1.

The notion of strong lifting compactness is introduced for completely regular Hausdorff spaces, and its structural properties, as well as its relationship to the strong lifting, to measure compactness, and to lifting compactness, are discussed. For metrizable locally convex spaces under their weak topology, strong lifting compactness is characterized by a list of conditions which are either measure theoretical or topological in their nature, and from which it can be seen that strong lifting compactness is the strong counterpart of measure compactness in that case.

We present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

In a recent paper Taylor and Tricot [10] introduced packing measures in ℝd. We modify their definition slightly to extend it to a general metric space. Our main concern is to show that in any complete separable metric space every analytic set of non-σ-finite h-packing measure contains disjoint compact subsets each of non-σ-finite measure. The corresponding problem for Hausdorff measures is discussed, but not completely resolved, in Rogers' book [7]. We also show that packing measure cannot be attained by taking the Hausdorff measure with respect to a different increasing function using another metric which generates the same topology. This means that the class of pacing measures is distinct from the class of Hausdorff measures.

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.

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].

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?

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.

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.