We provide a simple and direct proof of the completeness of the L1-space of any vector measure taking its values in the class of Fréchet spaces which do not contain a copy of the sequence space ω.

Iseki [11] defined a general notion of ergodicity suitable for functions ϕ: J → X where J is an arbitrary abelian semigroup and X is a Banach space. In this paper we develop the theory of such functions, showing in particular that it fits the general framework established by Eberlein [9] for ergodicity of semigroups of operators acting on X. Moreover, let A be a translation invariant closed subspace of the space of all bounded functions from J to X. We prove that if A contains the constant functions and ϕ is an ergodic function whose differences lie in A then ϕ ∈ A. This result has applications to spaces of sequences facilitating new proofs of theorems of Gelfand and Katznelson-Tzafriri [12]. We also obtain a decomposition for the space of ergodic vectors of a representation T: J → L(X) generalizing results known for the case J = Z+. Finally, when J is a subsemigroup of a locally compact abelian group G, we compare the Iseki integrals with the better known Cesàro integrals.

We show that a measure on ℝd is linearly rectifiable if, and only if, the lower l-density is positive and finite and agrees with the lower average l-density almost everywhere.

It is known that if A and B are nontriangular 2 × 2 non-negative integral matrices similar over the integers and –tr A ≤det A, then A and B are strongly shift equivalent. Suppose that A and B are 2 × 2 non-negative integral matrices similar over the integers. In this article it is shown that if –2 tr A≤det A <– tr A and if | det A | is not a prime, then A and B are strongly shift equivalent.

The aim of this paper is to resolve Taylor's question concerning certain regularity conditions on a Borel measure. The proposed solution is given in the framework of Brown, Michon and Peyrière, and Olsen.

We provide a unified and simplified proof that for any partition of (0, 1] into sets that are measurable or have the property of Baire, one cell will contain an infinite sequence together with all of its sums (finite or infinite) without repetition. In fact any set which is large around 0 in the sense of measure or category will contain such a sequence. We show that sets with 0 as a density point have very rich structure. Call a sequence and its resulting all-sums set structured provided for each We show further that structured all-sums sets with positive measure are not partition regular even if one allows shifted all-sums sets. That is, we produce a two cell measurable partition of (0, 1 ] such that neither set contains a translate of any structured all-sums set with positive measure.

We describe measurable Hilbert sheaves as Hilbert space objects in a sheaf category constructed from a measure space. These are quite useful for the interpretation of the direct integral of Hilbert spaces as an indexed functor. We set up a framework to put this and similar constructions of operator theory on an indexed categorical footing.

We consider the Riesz product with a constant coefficient and odometer action over infinite product spaces. By studying the ratio set we can conclude the type of the above dynamical systems is III1.

Dealing with a problem posed by Kupka we give results concerning the permanence of the almost strong lifting property (respectively of the universal strong lifting property) under finite and countable products of topological probability spaces. As a basis we prove a theorem on the existence of liftings compatible with products for general probability spaces, and in addition we use this theorem for discussing finite products of lifting topologies.

Throughout this paper we assume that k is a given positive integer. As usual, B(x, r) denotes the closed ball with centre at x∈ℝk and radius r > 0. Let μ be a Radon measure on ℝk, that is, μ is locally finite and Borel regular. For s ≥ 0, the lower and upper s–dimensional densities of μ at x are denned respectively by

and

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let f Θ,D :I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = f Θ,D (X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

Let be a sequence of mutually disjoint open balls, with centres xj and corresponding radii aj, each contained in the closed unit ball in d-dimensional euclidean space, ℝd. Further we suppose, for simplicity, that the balls Bj are indexed so that aj≥aj+1. The set

obtained by removing, from the balls {Bj} is called the residual set. We say that the balls {Bj} constitute a packing of provided that λ(ℛ)=0, where λ denotes the d-dimensional Lebesgue measure. Thus it follows that henceforth denoted by c(d), whilst the packing restraint ensures that Larman [11] has noted that, under these circumstances, one also has .

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

For a compact group G, we compute the Kazhdan constants κ(G, G) obtained by taking G itself as a generating subset. We get κ(G, G) = if G is finite of order n, and κ(G, G) = if G is infinite.

Neveu's exchange formula relates the Palm probabilities with respect to two jointly stationary simple point processes. We give a new proof of the exchange formula by using a simple result from discrete time stationary stochastic processes.

We present a systematic method of approximating, to an arbitrary accuracy, a probability measure µ on x = [0,1]q , q 1, with invariant measures for iterated function systems by matching its moments. There are two novel features in our treatment. 1. An infinite set of fixed affine contraction maps on , w 2, · ·· }, subject to an ‘ϵ-contractivity' condition, is employed. Thus, only an optimization over the associated probabilities pi is required. 2. We prove a collage theorem for moments which reduces the moment matching problem to that of minimizing the collage distance between moment vectors. The minimization procedure is a standard quadratic programming problem in the pi which can be solved in a finite number of steps. Some numerical calculations for the approximation of measures on [0, 1] are presented.

Guided by analogy with Euler's spherical excess formula, we define a finite-additive functional on bounded convex polygons in ℝ2 (the Euler functional). Under certain smoothness assumptions, we find some sufficient conditions when this functional can be extended to a planar signed measure. A dual reformulation of these conditions leads to signed measures in the space of lines in ℝ2. In this way we obtain two sets of conditions which ensure that a segment function corresponds to a signed measure in the space of lines. The latter conditions are also necessary.

Packing measures have been introduced to complement the theory of Hausdorff measures in [13,14]. (For a new treatment see also [10, Chapter 5]. While Hausdorff measures are intimately connected to upper density estimates (see, e.g., [5,2.10.18]), the importance of packing measures stems from their connection to lower density estimates.

This paper is concerned with the geometry of a measure μ, and in particular with the relationship between various .s-dimensional densities of μ, the geometry of the support of μ and the question of whether s is an integer.

We show that there exists an open set H⊆[0, 1] × [0, 1] with λ2(H) = 1 such that for any ε > 0 there exists a set E satisfying and H contains the product set E × E but there is no set S with and S × S ⊆ H. Especially this property is verified for sets of the form H = where the sets Ei are independent and . The results of this paper answer questions of M. Laczkovich and are related to a paper of D. H. Fremlin.