On continuous image averaging of Borel measures

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.

Let m be a Radon measure on a Hausdorff topological space X.Corresponding to three kinds of outer measures, three kinds of m-negligible sets are considered.The main theorem states that in a metacompact space X each locally m-negligible set is m-negligible.For a Radon measure in a HausdorfT topological space X we distinguish three kinds of negligible sets corresponding to three kinds of outer measures: strictly negligible, negligible and locally negligible sets, respectively.It is shown that two classes of negligible sets coincide iff the corresponding outer measures coincide.Further, each locally negligible set is strictly negligible if X is Lindelf.We give two examples that both in locally compact separable spaces and in locally compact metrizable spaces a negligible set need not be strictly negligible.It is an unsolved problem whether a locally negligible set is always negligible, or equivalently, whether the corresponding outer measures are always the same.In this latter form the problem is due to Schwartz [5, p. 17].The main purpose of this note is to prove that in a metacompact space, and hence in a paracompact space, each locally negligible set is negligible.It will turn out that this is even true for compact inner regular Borel measures which are only locally a-finite.At least in this slightly more general situation it can be shown that the assumption " metacompact" cannot be replaced by "locally compact".By the compact inner regularity of Radon measures the necessity to distinguish several kinds of negligible sets arises with the infinity of the measures.That is, for finite, compact inner regular Borel measures the three kinds of negligible sets coincide.In the case of finite Borel measures (not necessarily compact inner regular) there are at most two kinds of negligible sets.The class of topological spaces in which these two kinds coincide for each finite or each finite regular Borel measure was considered by Gardner in [3] (see also [4]).Throughout this paper X is a Hausdorff topological space.(X), \}(X) and Si(X) denote, respectively, the families of all open, closed and compact subsets of X.The Borel a-algebra is denoted by %5(X).A Borel measure in X, i.e. a nonnegative, countably additive set function on 93 (X), is called locally finite if each point in X has a neighbourhood of finite measure.

The first order theory of the decomposition of measures with respect to dimension which has been developed by Kahane, Katznelson, Cutler, and others is extended through transfinite recursion to a ω 1 \omega _1 -order theory. Necessary and sufficient conditions for a finite regular Borel measure on [ 0 , d ] ω 1 [0,d]^{\omega _1} to be a ω 1 \omega _1 -order multispectrum for a finite Borel measure on R d \mathbb {R}^d is given.

Multicyclicity of unbounded normal operators and polynomial approximation in C

Let G be an LCA group, H a closed subgroup, Γ the dual group of G and μ be a regular finite non-negative Borel measure on Γ. Motivated by the problem of sampling and reconstruction of a stationary random process over G from the values of m different linear measurements on H, we give some necessary or sufficient conditions for the density of certain sets of functions in which arise in multichannel sampling. Extensions of these results to and the existence of unconditional convergent expansions are discussed.

Let G ∗ {G^ * } be the maximal convolution operator associated with a sequence of L 1 {L^1} kernels. We show that if G ∗ {G^ * } is of weak type ( 1 , q ) (1,q) , 1 ≤ q > ∞ 1 \leq q > \infty , over a subset N {\mathcal N} of M {\mathcal M} (the space of all finite positive Borel measures on R h {{\bf {R}}^h} endowed with the weak topology), then G ∗ {G^ * } is of weak type ( 1 , q ) (1,q) over the closed cone in M {\mathcal M} generated by N {\mathcal N} . As a particular case we obtain a well-known result by de Guzman.

We show that in metalindelf spaces certain questions about Radon measures cannot be decided within the Zermelo-Fraenkel set theory, including the axiom of choice.0. Introduction.All spaces in this paper will be Hausdorff.Let A be a space.By and Q we shall denote, respectively, the families of all open and compact subsets of X.The Borel o-algebra in X, i.e., the smallest a-algebra in X containing , will be denoted by < .The elements of 9> are called Borel sets.A Borel measure in A' is a measure p on % such that each x G X has a neighborhood U G % with p( U) <A Radon space is a space X in which each finite Borel measure is Radon.The present paper will be divided into two parts, each dealing with one of the following questions.(a) Let A' be a locally compact, hereditarily metalindelf space with the countable chain condition.Is X a Radon space?() Let A' be a metalindelf space, and let p be a a-finite Radon measure in X.Is pa.regular measure?We shall show that neither question can be decided within the usual axioms of set theory.The proofs are based on an alternate application of the continuum hypothesis, Martin's axiom and Ostaszewski's axiom A. In passing, we shall also answer a question of F. D. Tall (see [T, last paragraph in 6]).Namely, using the continuum hypothesis and +, we shall construct a locally compact, hereditarily metalindelf space X with the countable chain condition which is not weakly 0-refinable.

Suppose that G is a bounded simply connected domain on the plane with boundary Γ, z0 ∈ G, ω is the harmonic measure with respect to z0 on Γ, µ is a finite Borel measure with support supp(µ) \(\subseteq \) Γ, µa + µs is the decomposition of µ with respect to ω, and t is a positive real number. We solve the following problem: for what geometry of the domain G is the condition $$\int {\ln \left( {\tfrac{{d\mu _a }}{{d\omega }}} \right)} d\omega = - \infty $$ equivalent to the completeness of the polynomials in Lt(µ) or to the unboundedness of the calculating functional p →p (z0), where p is a polynomial in Lt(µ)? We study the relationship between the densities of the algebras of rational functions in Lt(µ) and C(Γ). For t=2, we obtain a sufficient criterion for the unboundedness of the calculating functional in the case of finite Borel measures with support of an arbitrary geometry.

Let be the Hardy class in the unit disc and a finite Borel measure in . Carleson's theorem describes conditions on under which the corresponding imbedding operator (the Carleson operator) is bounded. From this theorem follows a criterion for compactness of in terms of .This paper is devoted to further study of the Carleson operator. Almost sharp upper bounds on the singular numbers of are presented in terms of the intensity of . For measures whose support is a set of nonzero linear measure adjacent to the unit circle (and when certain other conditions), an asymptotic formula is obtained. A study is begun of measures whose support has just one point on the unit circle. A solution of a problem from the theory of rational approximation, posed by A. A. Gonchar, is also presented.Bibliography: 17 titles.

Let G be a reflexive topological group, and Gits group of characters, endowed with the compact open topology. We prove that the evaluation mapping from Gx G into the torus T is continuous if and only if G is locally compact. This is an analogue of a well-known theorem of Arens on admissible topologies on C(X) . DEFINITIONS AND REMARKS Let X, Y be topological spaces, and let Z be a subset of yX . A topology on Z is said to be admissible if the evaluation mapping from the product Z x X into Y, defined by w(f, x) = f(x), is continuous. Let (S, V) = {f E Z; f(S) C V}. The family {(S, V)}, where S runs over the collection of all compact subsets of X and V runs over a basis of open sets in Y, is a subbase for the compact open topology on Z. An admissible topology on Z must be finer than the compact open topology [8]. A result of Arens states that the existence of a coarsest admissible topology for the class of real continuous functions on a completely regular space X is equivalent to X being locally compact [1]. In this paper we are interested in reflexive topological groups. Answering in the negative a question of Megrelishvili [7], we will see that the evaluation map for those groups need not be continuous. In fact, as specified in the theorem, the continuity of the evaluation characterizes locally compact Hausdorff abelian groups among reflexive groups. We prove this result by means of convergence spaces. For an account of theory of convergence spaces the reader is referred to [3, 4]. We only give here needed definitions. Let G be a Hausdorff topological abelian group, and let FG be the set of all continuous homomorphisms from G into the unidimensional torus T. If addition is defined pointwise in FG, it becomes an abelian group. The continuous Received by the editors October 5, 1993 and, in revised form, April 25, 1994; the contents of this paper were presented to the First Ibero-American Conference on Topology and its Applications in Benicassim, Spain, on March 28-30, 1995. 1991 Mathematics Subject Classification. Primary 22A05.

Admissible Representations of Probability Measures

We extend the results obtained in [14] by introducing a new class of boundary value problems involving non-local dynamic boundary conditions. We focus on the problem to find a solution to a local problem on a domain $$\varOmega $$ Ω with non-local dynamic conditions on the boundary $$\partial \varOmega $$ ∂ Ω . Due to the pioneering nature of the present research, we propose here the apparently simple case of $$\varOmega =(0, \infty )$$ Ω = ( 0 , ∞ ) with boundary $$\{0\}$$ { 0 } of zero Lebesgue measure. Our results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. Moreover, in our view, we bring new light to dynamic boundary value problems and the probabilistic description of the associated models.

Asymptotics for varying discrete Sobolev orthogonal polynomials

Carleson measures on the generalized Hartogs triangles

Strong Asymptotic Behavior and Weak Convergence of Polynomials Orthogonal on an Arc of the Unit Circle