Haar Null Sets and the Consistent Reflection of Non-meagreness

Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$ , the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number.

We describe and characterize all homogeneous subsets of the Cantor set which are both an F"s and a GSo; it turns out that there are wx such spaces.1. Introduction.All spaces under discussion are separable and metrizable.The aim of this paper is to determine and characterize all homogeneous zero-dimensional absolute Borel sets of ambiguous class 2, i.e., all homogeneous Borel sets in the Cantor set C that are both an FaS (a countable intersection of a-compact sets) and a GSa (a countable union of topologically complete spaces).Among the absolute Borel sets of ambiguous class 2 are all spaces that are either a-compact or topologically complete (they are the absolute Borel sets of class 1); characterizations of all zero-dimensional homogeneous such spaces have been known since around 1930: in 1910 Brouwer [3] characterized the Cantor set C; in 1920 Sierpihski [22] characterized the space of rationals Q; and in 1928 Alexandroff and Urysohn [1] characterized the Cantor set minus a point, C \ { p}, the space of irrationals P, and Q X C. It is easily seen that, apart from the discrete spaces, these are the only homogeneous zero-dimensional absolute Borel sets of class 1.It was not until 1981 that the first homogeneous zero-dimensional absolute Borel set of ambiguous class 2 which is not of class 1 was characterized: in [16] van Mill considered products of Q, P, and C, noted that Q X P was not yet characterized, and filled in this gap; two other elements of this class, S and T, which are homogeneous complements of Q X P in the Cantor set, were characterized by van Mill [17] resp., van Douwen ([4]; see also [6]), and Q X S and Q X T followed in [6] by van Engelen and van Mill.In this paper we show that there are precisely Wj homogeneous Borel sets of ambiguous class 2 in C; it is unknown how many homogeneous Borel sets (of arbitrary class) there are.This paper is organized as follows: in 3 we define topological properties Pa, for a < Uf, and describe how, with each of those properties, a finite number of homogeneous Borel sets of ambiguous class 2 in the Cantor set can be associated. 4 and 5 provide us with the lemmas that are fundamental to the construction (for the results of 5 we heavily rely on a technique due to Saint-Raymond [21]); the actual construction is carried out in 6 (spaces associated with property ^a for some a < u) and 7 (spaces associated with property 3ia for some a > to).In 8 we show that C contains no other homogeneous Borel sets of ambiguous class 2 than those of 6 and 7. Finally, 9 lists some open questions.

We prove that the category M of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, M is an abelian category containing A as a full subcategory such that the inclusion functor A→M is exact and finitely continuous. Furthermore, M is uniquely characterized up to equivalence by the following universal property: for every abelian category B, a functor A→B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M→B. In particular, this provides a description of the left heart of A as a concrete category.We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R-modules, for a given Polish group or Polish ring R; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.

The class of Borel sets in a locally compact space is the --ring generated by the compact sets [3, p. 219], while the class of weakly Borel sets is the o--algebra generated by the closed sets [2]. The objectof this note is to show that any measure defined on the class of Borel sets may be extended to the class of weakly Borel sets in a simple and canonical way (Theorem 1). There is a useful application to regular Borel measures (Theorem 2) and, in particular, to Haar measure (Theorem 3). The first part of our discussion is valid for an arbitrary measure space (X, 8, ,) in the sense of Halmos [3, p. 73]. Recall that the sets E in 8 are called measurable. We say that a subset A of X is locally measurable (with respect to S) if its intersection with every measurable set is measurable, that is, if EO\A ES for every E in S. We write ,S, for the class of all locally measurable sets; it is easy to see that S, is a o--algebra containing S. Moreover, if Sx is regarded as a ring in the usual way, then 8 is an ideal in Sx. The measure , may be extended to a measure yx on Sx in the following way. For each measurable set E, let yE be the measure on Sx defined by the formula 1IE(A) =y(EG\A); the family of measures { E: EC-S } is increasingly directed in the obvious sense, and, defining

Havin-Mazya type uniqueness theorem for Dirichlet spaces

0. Introduction. Let G be a locally compact Abelian group, and G* its group of (continuous) characters. We write (y, x) to indicate the function on G*XG which is the value of the character yEG* at the point xEG, or, dually, the value of the character xGG at the point yEG*. Borel sets in G are taken to be the smallest o--algebra of sets containing all closed sets. We consider the set of all regular, countably additive, complex-valued, Borel measures on G. (For measuretheoretic terms not defined here, see [4].) For any such measure k, let | =sup E | 4(A,) j, the supremum being taken over all finite collections {A, }l of Borel subsets of G which are pairwise disjoint and whose union is G. We postulate that 11k11 be finite. Such measures are called here bounded Radon measures. The measure er such that eZ(A) = 1 or 0 as A does or does not contain the point xEG is a particularly simple bounded Radon measure. Let Go:(G) denote the linear space of all continuous complex-valued functions f on G such that for every e > 0, there is a compact subset A of G such that lf(x) I <e for xEEA. It is known [6] that a complex linear functional L on (Fo(G) which is bounded in the uniform norm has a unique representation as the integral over G with respect to a bounded Radon measure 45:

Computable topological abelian groups

In this paper, we primarily introduce the concepts of random measure-theoretic N-expansivity and random measure-theoretic cw-expansivity for a Borel probability measure, as well as the notions of random N-expansivity and random cw-expansivity for a random dynamical system. Subsequently, we prove that a random dynamical system is randomly N-expansive (res., randomly cw-expansive) if and only if every Borel probability measure is randomly measure-theoretic N-expansive (res., randomly measure-theoretic cw-expansive) of f and all Borel probability measures share a common random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic), namely, there is a measure random variable δ : Ω → ( 0 , + ∞ ) such that δ is a random measure-theoretic N-expansive characteristic (res., random measure-theoretic cw-expansive characteristic) of every Borel probability measure.

Admissible Representations of Probability Measures

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F ⊆ P(X) of size ϲ, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f : X → X with f−1[{x}] ∉ I for each x ∈ X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B ∉ I and a perfect set P ⊆ X for which the family {B+x : x ∈ P} is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) ⇒ (M) ⇒ (B) ⇒ not-ccc can hold. We build a σ-ideal on the Cantor group witnessing (M) &amp; ¬(D) (Section 2). A modified version of that σ-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for “nicely” defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp version's of (M) for invariant ideals on Polish groups are investigated (Section 6).

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single‐valued as well as for multi‐valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya‐Srivastava in order to prove a generalization of the Representation Theorem of Kreitz‐Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach‐Hausdorff‐Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Continuum-wise expansive measures

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

Capacities on a numerical model — Tigtness of capacities — Quasicontinuous functions — Tchebycheff inequalities — Redefinitions — Positive generalized functionals, their representation by a Borel measure — Equilibrium potentials — Continuity of capacities — Capacitability of Borel stes — Equilibrium measures — Measures of finite energy — Charge and capacity — Slim sets are null sets for all measures of finite energy — Invariance of capacities under a change of numerical model, precise Gaussian probability space — Quasi-sure analysis on an abstract Wiener space.