On the existence of a finite invariant measure

A necessary and sufficient condition is given for a Borel automorphism on a standard Borel space to admit an invariant probability measure.     

Representing probability measures using probabilistic processes

In the Type-2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words, under which each successive symbol is determined by a finite probabilistic choice, generate Borel probability measures on the represented space. Conversely, for several well-behaved types of space, every Borel probability measure is represented by a corresponding probabilistic process. Accordingly, we consider probabilistic processes as providing “probabilistic names” for Borel probability measures. We show that integration is computable with respect to the induced representation of measures.     

圆内亚纯函数与其微分多项式的公共Borel点

该文把A.P.Singh关于一类齐次微分多项式级的结果推广到更一般的微分多项式。并证明了：如果Q（f)≠0是圆内有限正级亚纯函数f的身长分多项式,则f^(k0Q(f)的Borel点必是f的Borel点,其中K0满足0≤K0≤min{K:f^(k)出现在Q（f)中}。     

Central limit theorem for complex measures

We prove two central limit theorems for real identically distribution random variables where the distribution is a complex-valued Borel measure . The results involve the weak convergence of the suitably scaledn-fold convolution of certain complex atomic or absolutely continuous measures of spectral radius 1 to ahypergaussian measure whose Fourier-Stieltjes transform is exp(–² for a positive integer . The proof uses a generalization of the method of characteristic functions. Counter-examples are given to rather more general statements that had appeared previously in the literature. This research arose in connection with problems related to general tauberian theorems on the line for complexvalued summability methods which are discussed at the end of the paper. Some interesting examples are given generalizing well-known theorems related to Euler and Borel summability.     

On the structure of the set bases of a degenerate point

Consider an extreme point (EP)x ⁰ of a convex polyhedron defined by a set of linear inequalities. If the basic solution corresponding tox ⁰ is degenerate,x ⁰ is called a degenerate EP. Corresponding tox ⁰, there are several bases. We will characterize the set of all bases associated withx ⁰, denoted byB ⁰. The setB ⁰ can be divided into two classes, (i) boundary bases and (ii) interior bases. For eachB ⁰, there is a corresponding undirected graphG ⁰, in which there exists a tree which connects all the boundary bases. Some other properties are investigated, and open questions for further research are listed, such as the connection between the structure ofG ⁰ and cycling (e.g., in linear programs).     

On Borel probability measures and noncooperative game theory

In this article, the well-known minimax theorems of Wald, Ville and von Neumann are generalized under weaker topological conditions on the payoff function f and/or extended to the larger set of the Borel probability measures, instead of the set of mixed strategies.     

Density of the set of positive proper minimal points in the set of minimal points

One of the important problems of vector optimization concerns the density of the set of positive proper minimal points in the set of minimal points. We use the concepts of dentable point and approximating cones to derive sufficient conditions guaranteeing that the set of minimal points is contained in the closure of the set of positive proper minimal points. The result can be applied to obtain a density result for the unit ball in 1 ^p , 1<p<+, which does not follow from any other well-known density theorem.The author would like to thank Professor W. T. Fu for helpful comments. Moreover, the author is grateful to Professor H. P. Benson and the referees for valuable remarks and suggestions concerning a previous draft of this paper.     

Density of periodic points, invariant measures and almost equicontinuous points of cellular automata

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of such automata converges in Cesàro mean to an invariant measure μ_c. Moreover the dynamical system (cellular automaton F, invariant measure μ_c) has still the μ_c-almost equicontinuity property and the set of periodic points is dense in the topological support of the measure μ_c. We also show that the density of periodic points in the topological support of a measure μ occurs for each μ-almost equicontinuous cellular automaton when μ is an invariant and shift ergodic measure. Finally using most of these results we give a non-trivial example of a couple (μ-equicontinuous cellular automaton F, shift and F-invariant measure μ) such that the restriction of F to the topological support of μ has no equicontinuous points.     

On the integration of Radon measures and set functions

A theory of integration is described which subsumes as special cases Bourbaki's integration of Radon measures and Caratheodory's integration of set functions.Supported partly by NSF Grant GP-20541     

Invariant measures for interval maps with critical points and singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.     

Classifying countable Boolean terms

We deal with the Borel and difference hierarchies in the space P of all subsets of endowed with the Scott topology. (The spaces P and 2 coincide set-theoretically but differ topologically.) We look at the Wadge reducibility in P. The results obtained are applied to the problem of characterizing ₁-terms t which satisfy C = t( ₁ ⁰ ) for a given Borel-Wadge class C. We give its solution for some levels of the Wadge hierarchy, in particular, all levels of the Hausdorff difference hierarchy. Finally, we come up with a discussion of some relevant facts and open questions.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 173–197, March–April, 2005.     

On the stereological estimation of reduced moment measures

Recently, a new set of fundamental stereological formulae based on isotropically oriented probes through fixed points have been derived, the so-called nucleator estimation principle (cf. Jensen and Gundersen (1989, J. Microsc., 153, 249–267)). In the present paper, it is shown how a model-based version of these formulae leads to stereological estimators of reduced moment measures of stationary and isotropic random sets in ⁿ .     

On the set of limit points of the partial sums of series rearranged by a given divergent permutation

We give a new characterization of divergent permutations. We prove that for any divergent permutation p, any closed interval I of (the 2-point compactification of ) and any real number sI, there exists a series ∑a_n of real terms convergent to s such that I=σa_p(n) (where σa_p(n) denotes the set of limit points of the partial sums of the series ∑a_p(n)). We determine permutations p of for which there exists a conditionally convergent series ∑a_n such that ∑a_p(n)=+∞. If the permutation p of possesses the last property then we prove that for any and there exists a series ∑a_n convergent to α and such that σa_p(n)=[β,+∞]. We show that for any countable family P of divergent permutations there exist conditionally convergent series ∑a_n and ∑b_n such that any series of the form ∑a_p(n) with pP is convergent to the sum of ∑a_n, while for every pP.     

On spaces of p-adic vector measures

LetX be a Hausdorff zero-dimensional topological space,K(X) the algebra of all clopen subsets of X, E a Hausdorff locally convex space over a non-Archimedean valued field and C _b (X) the space of all bounded continuous -valued functions on X. The space M(K(X),E), of all bounded finitely-additive measures m: K(X) → E, is investigated. If we equip C _b (X) with the topologies β _o , β, β _u , τ _b or β _ob , it is shown that, for E (compete, the corresponding spaces of continuous linear operators from C _b (X) to E (are algebraically isomorphic to certain subspaces of M(K(X),E). The text was submitted by the author in English.     

On the maximal operator associated with certain rotational invariant measures

The aim of this work is to investigate the integrability properties of the maximal operator Mu,associated with a non-doubling measure μ defined on Rn. We start by establishing for a wide class of radial and increasing measures μ that Mu is bounded on all the spaces Lu^p（R^n）,P〉1.Also,we show that there is a radial and increasing measure p for which Mμ does not map Lμ^p（R^n） into weak Lμ^p（R^n）,1≤p〈∞.     

On the relationship between fixed points and iteration in admissible set theory without foundation

In this article we show how to use the result in Jäger and Probst [7] to adapt the technique of pseudo-hierarchies and its use in Avigad [1] to subsystems of set theory without foundation. We prove that the theory KPi⁰ of admissible sets without foundation, extended by the principle (Σ-FP), asserting the existence of fixed points of monotone Σ operators, has the same proof-theoretic ordinal as KPi⁰ extended by the principle (Σ-TR), that allows to iterate Σ operations along ordinals. By Jäger and Probst [6] we conclude that the metapredicative Mahlo ordinal φω00 is also the ordinal of KPi⁰+(Σ-FP). Hence the relationship between fixed points and iteration persists in the framework of set theory without foundation.The author is supported by the Swiss National Science Foundation.     

On a Marinacci uniqueness theorem for measures

We prove a uniqueness theorem for measures on D-posets.