On Uncountable Unions and Intersections of Measurable Sets

We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a good additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are considered in detail, and it is shown that the Lebesgue measure and the Baire property differ essentially in this aspect. Another difference between measure and category is illustrated in the case of some uncountable intersections of sets of full measure (comeager sets, respectively). We also discuss a topological form of the Vitali covering theorem, in connection with the Baire property of uncountable unions of certain sets.     

Computability of measurable sets via effective topologies

We investigate in the frame of TTE the computability of functions of the measurable sets from an infinite computable measure space such as the measure and the four kinds of set operations. We first present a series of undecidability and incomputability results about measurable sets. Then we construct several examples of computable topological spaces from the abstract infinite computable measure space, and analyze the computability of the considered functions via respectively each of the standard representations of the computable topological spaces constructed. The authors are supported by grants of NSFC and DFG.     

A Non-commutative Yosida–Hewitt Theorem and Convex Sets of Measurable Operators Closed Locally in Measure

We present a non-commutative extension of the classical Yosida–Hewitt decomposition of a finitely additive measure into its σ-additive and singular parts. Several applications are given to the characterisation of bounded convex sets in Banach spaces of measurable operators which are closed locally in measure.     

Finite-tight sets

We introduce two notions of tightness for a set of measurable functions — the finite-tightness and the Jordan finite-tightness with the aim to extend certain compactness results (as biting lemma or Saadoune-Valadier’s theorem of stable compactness) to the unbounded case. These compactness conditions highlight their utility when we look for some alternatives to Rellich-Kondrachov theorem or relaxed lower semicontinuity of multiple integrals. Finite-tightness locates the great growths of a set of measurable mappings on a finite family of sets of small measure. In the Euclidean case, the Jordan finite-tight sets form a subclass of finite-tight sets for which the finite family of sets of small measure is composed by d-dimensional intervals. The main result affirms that each tight set H ⊆ W ^1,1 for which the set of the gradients ∇H is a Jordan finite-tight set is relatively compact in measure. This result offers very good conditions to use fiber product lemma for obtaining a relaxed lower semicontinuity condition.      

Measurability and the abstract Baire property

Within an abstract theory of point sets the author has successfully unified a substantial number of the analogous theorems concerning Lebesgue measure and Baire category. It has been shown that the Lebesgue measurable sets coincide with the sets having the abstract Baire property with respect to the family of all closed sets of positive Lebesgue measure and the question was raised in [2] whether the sets measurable with respect to certain Hausdorff measures were the same as the sets having the abstract Baire property with respect to the family of all closed sets of positive Hausdorff measure. In this article we establish a general theorem which, under the assumption of the continuum hypothesis, gives an affirmative answer to this question.     

The logic of integration

We develop a model theoretic framework for studying algebraic structures equipped with a measure. The real line is used as a value space and its usual arithmetical operations as connectives. Integration is used as a quantifier. We extend some basic results of pure model theory to this context and characterize measurable sets in terms of zero-sets of formulas.      

Graham-Rothschild parameter words and measurable partitions

Generalizing earlier results of Moran and Strauss (Mathematika 27, 1980, 213–244) and of Carlson and Simpson (Adv. in Math. 53, 1984, 265–290) it was shown in Prömel and Voigts (Trans. Amer. Math. Soc. 291, 1985, 198–201) that Baire sets ofk-parameter words are Ramsey. Motivated by the duality between category and measure, we investigate in this paper measurable sets ofk parameter words. We show that measurable sets of ascendingk-parameter words are Ramsey, whereas in general measurable sets ofk-parameter words fail to be Ramsey.     

Statistical convergence and ideal convergence for sequences of functions

Let I⊂P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M,μ), we obtain a statistical version of the Egorov theorem (when μ(X)<∞). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I-convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.     

On the volume of convex hulls of sets on spheres

We prove that for a measurable subset of S ^n–1 with fixed Haar measure, the volume of its convex hull is minimized for a cap (i.e. a ball with respect to the geodesic measure). We solve a similar problem for symmetric sets and n=2, 3. As a consequence, we deduce a result concerning Gaussian measures of dilatations of convex, symmetric sets in R ² and R ³.Partially supported by KBN (Poland), Grant No. 2 1094 91 01.     

Statistical maps: A categorical approach

In probability theory, each random variable f can be viewed as channel through which the probability p of the original probability space is transported to the distribution p _f , a probability measure on the real Borel sets. In the realm of fuzzy probability theory, fuzzy probability measures (equivalently states) are transported via statistical maps (equivalently, fuzzy random variables, operational random variables, Markov kernels, observables). We deal with categorical aspects of the transportation of (fuzzy) probability measures on one measurable space into probability measures on another measurable spaces. A key role is played by D-posets (equivalently effect algebras) of fuzzy sets. Supported by VEGA 1/2002/06.     

Best approximation properties in spaces of measurable functions

We research proximinality of μ‐sequentially compact sets and μ‐compact sets in measurable function spaces. Next we show a correspondence between the Kadec–Klee property for convergence in measure and μ‐compactness of the sets in Banach function spaces. Also the property S is investigated in Fréchet spaces and employed to provide the Kadec–Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fréchet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one‐dimensional subspace in sequence Lorentz spaces .     

List multicolorings of graphs with measurable sets

In an ordinary list multicoloring of a graph, the vertices are “colored” with subsets of pre‐assigned finite sets (called “lists”) in such a way that adjacent vertices are colored with disjoint sets. Here we consider the analog of such colorings in which the lists are measurable sets from an arbitrary atomless, semifinite measure space, and the color sets are to have prescribed measures rather than prescribed cardinalities. We adapt a proof technique of Bollobás and Varopoulos to prove an analog of one of the major theorems about ordinary list multicolorings, a generalization and extension of Hall's marriage theorem, due to Cropper, Gyárfás, and Lehel. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 179–193, 2007     

Measurable supports,reducible spaces and the structure of the optimal σ-field in unbiased estimation

This paper generalizes a result bySapozhnikov relating the structure of the optimal -field in unbiased estimation to the reducibility of certain linear subspaces of estimators. For this purpose, the concept of a measurable support of (uncountably infinite) sets consisting of random variables is introduced, which could be of general interest. It turns out, that for a wide class of measure spaces, measurable supports of any subsets of measurable functions exist.     

The gonality of smooth curves with plane models

The Hausdorff measure with fractional index is used in order to define a functional on measurable sets of the plane. A fractal set, constructed using the well-known Von Koch set, is involved in the definition. This functional is proved to arise as the limit of a sequence of classical functionals defined on sets of finite perimeter. Thus it is shown that a natural extension of the ordinary functionals of the calculus of variations leads both to fractal sets and to the fractional Hausdorff measure.     

Fuzzy probability measures

In [4] Höhle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see [6]).     

Upper integral and its geometric meaning

The Hahn definition of the integral is recalled, the requirement of measurability of the integrand omitted. Both the upper and lower integrals comply with this definition and so does any measurable function between them. The outer product measure of the hypograph of a nonnegative bounded nonmeasurable function is equal to the upper integral which is equal to one of the Fan integrals. The outer measure of the graph of a bounded nonmeasurable function is equal to the difference between the upper and lower integrals. A norm for not necessarily measurable functions is defined with the upper integral. The linear space with this norm is complete. The convergence in this space implies the convergence in outer measure. The distance as an outer measure of the symmetric difference of two sets gives us a complete metric space of classes of subsets.      

判别函数可积性的一个新的重要命题

在抽象测度空间中,用可测集EK去逼近集E的办法,从函数f在E上的可测性去推f在E上的可积性,是判别函数可积性的一个新的重要命题,但[2]在证明这一命题时有误.本文作了更正,并从距离空间中的积分推广到抽象测度空间中的积分.     

Extension of an abstract attainability problem with the use of the stone representation space

We consider an attainability problem in a complete metric space on values of an objective operator h. We assume that the latter admits a uniform approximation by mappings which are tier with respect to a given measurable space with an algebra of sets. Let asymptotic-type constraints be defined as a nonempty family of sets in this measurable space. We treat ultrafilters of the measurable space as generalized elements; we equip this space of ultrafilters with a topology of a zero-dimensional compact (the Stone representation space). On this base we construct a correct extension of the initial problem, realizing the set of attraction in the form of a continuous image of the compact of feasible generalized elements. Generalizing the objective operator, we use the limit with respect to ultrafilters of the measurable space. This provides the continuity of the generalized version of h understood as a mapping of the zero-dimensional compact into the topological space metrizable with a total metric.     

基于模糊值Choquet积分定义集函数的S性与PGP性

针对模糊测度空间上已建立的模糊值Choquet积分,将这种积分整体看成可测空间上取值于模糊数的集函数,当模糊测度满足一般S性和PGP性时,研究了这种模糊值集函数所保持的遗传性质.     

Note on the Permutations which Preserve Buck’s Measure Density

In the first part some conditions under which a permutation preserves Buck’s measure density, are derived. In the second part is constructed a countable system of permutations which has the ergodic property on the system of Buck’s measurable sets. Received: March 27, 2007., Accepted: August 25, 2008.