Borel Sets Research Articles

Analogs of the Lebesgue Measure in Spaces of Sequences and Classes of Functions Integrable with Respect to These Measures

We examine translation-invariant measures on Banach spaces lp, where p ∈ [1,∞]. We construct analogs of the Lebesgue measure on Borel σ-algebras generated by the topology of pointwise convergence (σ-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not σ-finite. We also study spaces of functions integrable with respect to measures constructed and prove that these spaces are not separable. We consider various dense subspaces in spaces of functions that are integrable with respect to a translation-invariant measure. We specify spaces of continuous functions, which are dense in the functional spaces considered. We discuss Borel σ-algebras corresponding to various topologies in the spaces lp, where p ∈ [1,∞]. For p ∈ [1,∞), we prove the coincidence of Borel σ-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel σ-algebra corresponding to the topology of pointwise convergence. We also verify that the space l∞ does not possess similar properties.

Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations

We establish a connection between random set theory and Boolean valued analysis by showing that random Borel sets, random Borel functions, and Markov kernels are respectively represented by Borel sets, Borel functions, and Borel probability measures in a Boolean valued model. This enables a Boolean valued transfer principle to obtain random set analogues of available theorems. As an application, we establish a Boolean valued transfer principle for large deviations theory, which allows for the systematic interpretation of results in large deviations theory as versions for Markov kernels. By means of this method, we prove versions of Varadhan and Bryc theorems, and a conditional version of Cramér theorem.

On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates

We show that, for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (sub-Gaussian estimates) imply the singularity of the energy measures with respect to the symmetric measure, verifying a conjecture by M. T. Barlow in (Contemp. Math. 338 (2003) 11–40). We also prove that in the contrary case of walk dimension two, that is, where full off-diagonal Gaussian estimates of the heat kernel hold, the symmetric measure and the energy measures are mutually absolutely continuous in the sense that a Borel subset of the state space has measure zero for the symmetric measure if and only if it has measure zero for the energy measures of all functions in the domain of the Dirichlet form.

Mass Transference Principle: From balls to arbitrary shapes: Measure theory

Following a work of Koivusalo and Rams, by a further generalization of the singular value function, we extend the Mass Transference Principle set up by Beresnevich and Velani to lim sup sets generated by open sets of arbitrary shapes. For a Borel set E⊂Rd with d≥1 and a dimension function f, define a generalized singular value function asφf(E):=supμ∈M(E)⁡infx∈E⁡infr>0⁡f(r)μ(E∩B(x,r)), where M(E) represents the collection of all Borel probability measures supported on E and B(x,r) denotes the ball of center x and radius r. Let {Bi}i∈N be a sequence of balls in [0,1]d with radius rBi→0 as i→∞. Assume that λ(lim supi→∞Bi)=1, where λ is the Lebesgue measure in Rd. Let {Ei}i∈N be a sequence of open sets such that Ei⊂Bi and limi→∞⁡λ(Bi)λ(Ei)=∞. Let f be a dimension function such that f(r)/rd is monotonically decreasing and limr→0⁡f(r)rd=∞ and φf(Ei)≥λ(Bi) for each pair (Ei,Bi). Then, for any ball B in [0,1]dHf(B∩lim supi→∞Ei)=Hf(B).

A Q-WADGE HIERARCHY IN QUASI-POLISH SPACES

AbstractThe Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff–Kuratowski (HK)-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

On the equality of two-variable general functional means

Given two functions f,g:Irightarrow mathbb {R} and a probability measure mu on the Borel subsets of [0, 1], the two-variable mean M_{f,g;mu }:I^2rightarrow I is defined by Mf,g;μ(x,y):=(fg)-1∫01f(tx+(1-t)y)dμ(t)∫01g(tx+(1-t)y)dμ(t)(x,y∈I).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} M_{f,g;\\mu }(x,y) :=\\bigg (\\frac{f}{g}\\bigg )^{-1}\\left( \\frac{\\int _0^1 f\\big (tx+(1-t)y\\big )d\\mu (t)}{\\int _0^1 g\\big (tx+(1-t)y\\big )d\\mu (t)}\\right) \\quad (x,y\\in I). \\end{aligned}$$\\end{document}This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure mu , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which Mf,g;μ(x,y)=MF,G;μ(x,y)(x,y∈I)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} M_{f,g;\\mu }(x,y)=M_{F,G;\\mu }(x,y) \\quad (x,y\\in I) \\end{aligned}$$\\end{document}holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means.

Playing with Mathias

We present a series of axioms that describe the combinatorial core of the Mathias model (countable support iteration of ω2 Mathias reals over the Continuum Hypothesis). Our axioms are formulated in terms of games with Borel sets and functions and are strong enough to imply most of the propositions usually proved by means of the iterated Mathias forcing.

Equivalence of codes for countable sets of reals

AbstractA set $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ is universal for countable subsets of ${\mathbb {R}}$ if and only if for all $x \in {\mathbb {R}}$ , the section $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ is countable and for all countable sets $A \subseteq {\mathbb {R}}$ , there is an $x \in {\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\mathbb {R}}$ by $x_0 \ E_U \ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\mathbf {\Sigma }_3^1$ -generic absoluteness, then for all $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ that are $\mathbf {\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .

Some results on random unimodular lattices

Let $n \in \mathbb{Z}_{\geq 3}.$ Given any Borel subset $A$ of $\mathbb{R}^n$ with finite and nonzero measure, we prove that the probability that the set of primitive points of a random full-rank unimodular lattice in $\mathbb{R}^n$ does not contain any $\mathbb{R}$-linearly independent subset of $A$ of cardinality $(n-2)$ is bounded from above by a constant multiple, which depends only on $n$, of $\left(\mathrm{vol}(A)\right)^{-1}.$ This generalizes a result that is jointly due to J. S. Athreya and G. A. Margulis (see \cite[Theorem 2.2]{Log}). We also generalize independent results of C. A. Rogers (see \cite[Theorem 6]{MeanRog}) and W. M. Schmidt (see \cite[Theorem 1]{Metrical}) about primitive lattice points of random lattices to the case of primitive tuples of rank less than $\frac{n}{2}.$ In addition to the work of the authors who were just mentioned, a crucial element of this present paper is the usage of a rearrangement inequality due to Brascamp\textendash Lieb\textendash Luttinger (see \cite[Theorem 3.4]{BLL}).

The isomorphism class of the shift map

The shift map, denoted σ, is the self-homeomorphism of ▪ induced by the successor function n↦n+1 on ω. We prove that the isomorphism classes of σ and σ−1 cannot be separated by a Borel set in H(ω⁎), the space of all self-homeomorphisms of ω⁎ equipped with the compact-open topology.Van Douwen proved it is consistent for σ and σ−1 not to be isomorphic. Whether it is also consistent for them to be isomorphic is an open problem. The theorem stated above can be thought of as a counterpoint to van Douwen's result: while σ and σ−1 may not be isomorphic, there is no simple topological property that distinguishes them.As a relatively straightforward consequence of the main theorem, we deduce that OCA+MA implies the set of continuous images of σ fails to be Borel in H(ω⁎). (Here a “continuous image” of σ is meant in the sense of topological dynamics: any h∈H(ω⁎) such that q∘σ=h∘q for some continuous surjection q:ω⁎→ω⁎.) This contrasts starkly with a recent theorem of the author showing that under CH, the continuous images of σ form a closed subset of H(ω⁎).

A probabilistic Takens theorem

Let be a Borel set, μ a Borel probability measure on X and T : X → X a locally Lipschitz and injective map. Fix strictly greater than the (Hausdorff) dimension of X and assume that the set of p-periodic points of T has dimension smaller than p for p = 1, …, k − 1. We prove that for a typical polynomial perturbation of a given locally Lipschitz function , the k-delay coordinate map is injective on a set of full μ-measure. This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, Bölcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required coordinates from 2 dim X to dim X and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results.

ON GILP’S GROUP-THEORETIC APPROACH TO FALCONER’S DISTANCE PROBLEM

AbstractIn this paper, we follow and extend a group-theoretic method introduced by Greenleaf–Iosevich–Liu–Palsson (GILP) to study finite points configurations spanned by Borel sets in $\mathbb{R}^n,n\geq 2,n\in\mathbb{N}.$ We remove a technical continuity condition in a GILP’s theorem in [Revista Mat. Iberoamer31 (2015), 799–810]. This allows us to extend the Wolff–Erdogan dimension bound for distance sets to finite points configurations with k points for $k\in\{2,\dots,n+1\}$ forming a $(k-1)$ -simplex.

On numbers of observations in random regions determined by records

Near-record observations are the ones that occur between successive record times and within a fixed distance of the current record value. In this paper, the concept of near-record observations is generalized to the notion of observations that fall into a random region determined by a given record and a Borel set. Description of the distribution of the number of such observations is provided, and asymptotic behaviour of this number is investigated. In addition, some applications of the new results to derive exact and asymptotic properties of inter-record times and of numbers of repetitions of records are presented.

Spectral Resolutions and Quantum Observables

An n-dimensional quantum observable in quantum structures is a kind of a σ-homomorphism defined on the Borel σ-algebra of $\mathbb R^{n}$ with values in a monotone σ-complete effect algebra or in a σ-complete MV-algebra. It defines an n-dimensional spectral resolution that is a mapping from $\mathbb R^{n}$ into the quantum structure which is a monotone, left-continuous mapping with non-negative increments and which is going to 0 if one variable goes to $-\infty $ and it goes to 1 if all variables go to $+\infty $ . The basic question is to show when an n-dimensional spectral resolution entails an n-dimensional quantum observable. We show cases when this is possible and we apply the result to existence of three different kinds of joint observables.

Coarse dimension and definable sets in expansions of the ordered real vector space

Let E⊆R. Suppose there is an s>0 such that |{k∈Z,−m≤k≤m−1:[k,k+1]∩E≠∅}|≥ms for all sufficiently large m∈N. Then there is an n∈N and a linear T:Rn→R such that T(En) is dense. As a corollary, we show that if E is in addition nowhere dense, then (R,<,+,0,(x↦λx)λ∈R,E) defines every bounded Borel subset of every Rn.

Ramsey theory without pigeonhole principle and the adversarial Ramsey principle

We develop a general framework for infinite-dimensional Ramsey theory with and without pigeonhole principle, inspired by Gowers’ Ramsey-type theorem for block sequences in Banach spaces and by its exact version proved by Rosendal. In this framework, we prove the adversarial Ramsey principle for Borel sets, a result conjectured by Rosendal that generalizes at the same time his version of Gowers’ theorem and Borel determinacy of games on integers.

Spaces with fragmentable open sets

The object of the paper are the regular topological spaces X in which there exists a metric d related to the topology in the following way: for every nonempty open subset U of X and for every ε>0 there exists a nonempty open subset V of U with d-diameter less than ε. It is shown that such a space X is pseudo-almost Čech complete if, and only if, it contains a dense completely metrizable subset (X is pseudo-almost Čech complete if it is a subset of some pseudocompact space Y and contains as a dense subset some Gδ-subset of Y).A large class of spaces L is described (containing all Borel subsets of compact spaces, all pseudocompact spaces and all p-spaces of Arhangel'skii) such that, if X belongs to L and admits a metric d with the above property, then X is “metrizable up to a first Baire category subset”. I.e. X is the union of two sets X1 and X2 where X1 is of the first Baire category in X and X2 is metrizable. We provide also different characterizations of the class L.

Two-Dimensional Observables and Spectral Resolutions

A two-dimensional observable is a special kind of a σ-homomorphism defined on the Borel σ-algebra of the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra. A two-dimensional spectral resolution is a mapping defined on the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra which has properties similar to a two-dimensional distribution function in probability theory. We show that there is a one-to-one correspondence between two-dimensional observables and two-dimensional spectral resolutions defined on a σ-complete MV-algebras as well as on the monotone σ-complete effect algebras with the Riesz decomposition property. The result is applied to the existence of a joint two-dimensional observable of two one-dimensional observables.

A Lower Density Operator for the Borel Algebra

We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the sigma -ideal of countable sets, for an uncountable Polish space, is equivalent to CH. One of the implications is known (due to K. Musiał) and the remaining implication is derived from a general abstract result dealing with the negation of GCH. We observe that there is no lower density Borel operator with respect to the sigma -ideal of countable sets, whose range is of bounded level in the Borel hierarchy.

Existence of an invariant measure on a topological quasigroup

In this article left or right invariant measures and functionals on topological unital quasigroups are investigated. It is proved that if under rather general conditions a nontrivial left invariant measure exists on a Borel σ-algebra of a topological unital quasigroup G, then G is locally compact.