There are $$2^{\mathfrak {c}}$$ Quasicontinuous Non Borel Functions on Uncountable Polish Space

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Ľubica Holá
Keyword(s):  
Polish Space ◽  
Borel Functions

Measurable chromatic numbers

2008 ◽  
Vol 73 (4) ◽  
pp. 1139-1157 ◽  
Author(s):  
Benjamin D. Miller
Keyword(s):  
Chromatic Number ◽  
Equivalence Relation ◽  
Initial Segment ◽  
Polish Space ◽  
Borel Function ◽  
Chromatic Numbers ◽  
Unilateral Shift ◽  
Dichotomy Theorem ◽  
Borel Functions ◽  
Measurable Chromatic Number

AbstractWe show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [ℕ]ℕ, although its Borel chromatic number is ℵ0. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In particular, this holds for “the” minimum analytic graph with uncountable Borel (and Baire measurable) chromatic number. In contrast, we show that for all κ Є6 (2, 3…..ℵ0, c), there is a treeing of E0 with Borel and Baire measurable chromatic number κ. Finally, we use a Glimm–Effros style dichotomy theorem to show that every basis for a non-empty initial segment of the class of graphs of Borel functions of Borel chromatic number at least three contains a copy of (ℝ<ℕ, ⊇).

scholarly journals Turing degrees in Polish spaces and decomposability of Borel functions

2020 ◽  
Vol 21 (01) ◽  
pp. 2050021
Author(s):  
Vassilios Gregoriades ◽  
Takayuki Kihara ◽  
Keng Meng Ng
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Recursion Theory ◽  
Descriptive Set Theory ◽  
Analytic Subset ◽  
Polish Spaces ◽  
Negative Results ◽  
Borel Functions ◽  
Important Open Problem ◽  
Descriptive Set

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture.

A quasi-order on continuous functions

2013 ◽  
Vol 78 (2) ◽  
pp. 633-648 ◽  
Author(s):  
Raphaël Carroy
Keyword(s):  
Polish Space ◽  
Continuous Functions ◽  
Borel Sets ◽  
Borel Functions ◽  
Quasi Order

AbstractWe define a quasi-order on Borel functions from a zero-dimensional Polish space into another that both refines the order induced by the Baire hierarchy of functions and generalises the embeddability order on Borel sets. We study the properties of this quasi-order on continuous functions, and we prove that the closed subsets of a zero-dimensional Polish space are well-quasi-ordered by bi-continuous embeddability.

scholarly journals A Central Limit Theorem for Predictive Distributions

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3211
Author(s):  
Patrizia Berti ◽  
Luca Pratelli ◽  
Pietro Rigo
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Polish Space ◽  
Borel Subset ◽  
Covariance Structure ◽  
Predictive Distribution ◽  
Gaussian Kernel ◽  
Finite Dimensional ◽  
Borel Functions ◽  
Random Probability

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions f:S→R. Let an(·)=P(Xn+1∈·∣X1,…,Xn) be the n-th predictive distribution corresponding to a sequence (Xn) of S-valued random variables. If (Xn) is conditionally identically distributed, there is a random probability measure μ on S such that ∫fdan⟶a.s.∫fdμ for all f∈F. Define Dn(f)=dn∫fdan−∫fdμ for all f∈F, where dn>0 is a constant. In this note, it is shown that, under some conditions on (Xn) and with a suitable choice of dn, the finite dimensional distributions of the process Dn=Dn(f):f∈F stably converge to a Gaussian kernel with a known covariance structure. In addition, Eφ(Dn(f))∣X1,…,Xn converges in probability for all f∈F and φ∈Cb(R).

Topological Completeness of Function Spaces Arising in the Hausdorff Approximation of Functions

1992 ◽  
Vol 35 (4) ◽  
pp. 439-448 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Approximation Theory ◽  
Metric Space ◽  
Polish Space ◽  
Complete Metric Space ◽  
Approximation Of Functions ◽  
Lower Envelopes ◽  
Constructive Approximation ◽  
Hausdorff Approximation ◽  
Set Of Points ◽  
Completely Metrizable

AbstractLet X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

Regulacja górnictwa kosmicznego w polskiej ustawie o działalności kosmicznej

2021 ◽  
pp. 171-186
Author(s):  
Mariusz Tomasz Kłoda ◽  
Katarzyna Malinowska ◽  
Bartosz Malinowski ◽  
Małgorzata Polkowska
Keyword(s):  
Polish Space ◽  
Eu Law ◽  
International Space ◽  
National Space ◽  
Space Activity ◽  
Space Law ◽  
The Usa ◽  
Polish Law ◽  
Space Activities ◽  
Do So

Work on the content of the law on space activities has been going in Poland for several years. So far, the drafters have not directly referred to the issue of space mining in the content of the proposed legal act. In this context, it is worth asking whether it is valuable and permissible, in terms of international space law and EU law, to regulate in the future (Polish) law on space activity the matter of prospecting, acquiring and using space resources, i.e. so-called space mining. If space mining were regulated in the Polish space law, Poland would not be the first country to do so. The discussed issues have already been regulated in the national space legislation of the USA, Luxemburg, UAE and Japan. This paper will analyze the issues of space mining as expressed in the current drafts of the Polish space law and foreign space legislation, of space mining as a means of achieving various goals and of the compatibility of space mining with international space law and EU law.

scholarly journals Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2845
Author(s):  
Sandra Fortini ◽  
Sonia Petrone ◽  
Hristo Sariev
Keyword(s):  
Polish Space ◽  
Asymptotic Properties ◽  
Transition Kernel ◽  
Urn Model ◽  
Additive Structure ◽  
Predictive Distributions ◽  
Pólya Urn ◽  
Random Weights ◽  
Random Transition ◽  
Polya Urn

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

Proof of the antithetic-variates theorem for unbounded functions

1979 ◽  
Vol 86 (3) ◽  
pp. 477-479 ◽  
Author(s):  
James R. Wilson
Keyword(s):  
Marginal Distribution ◽  
Functional Dependence ◽  
Finite Variance ◽  
Response Functions ◽  
Unbiased Estimators ◽  
Monte Carlo Techniques ◽  
Joint Distributions ◽  
Antithetic Variates ◽  
Borel Functions ◽  
Image Position

The method of antithetic variates introduced by Hammersley and Morton (2) is one of the most widely used Monte Carlo techniques for estimating an unknown parameter θ. The basis for this method was established by Hammersley and Mauldon(l).in the case of unbiased estimators with the formwhere each of the variates ξj is required to have a uniform marginal distribution over the unit interval [0,1]. By assuming that n = 2 and that the gj are bounded Borel functions, Hammersley and Mauldon showed that the greatest lower bound of var (t) over all admissible joint distributions for the variates ξj can be approached simply by arranging an appropriate strict functional dependence between the ξj. Handscomb(3) extended this result to the case of n > 2 bounded antithetic variates gj(ξj). In many experiments involving distribution sampling or the simulation of some stochastic process over time, the response functions gry are unbounded. This paper further extends the antithetic-variates theorem to include the case of n ≥ 2 unbounded antithetic variates gj(ξj) each with finite variance.

On the geometric ergodicity for a generalized IFS with probabilities

2021 ◽  
pp. 2150051
Author(s):  
Grzegorz Guzik ◽  
Rafał Kapica
Keyword(s):  
Polish Space ◽  
Iterated Function Systems ◽  
Wasserstein Distance ◽  
Multiple Delays ◽  
Time Behavior ◽  
Limiting Behavior ◽  
Borel Measures ◽  
Long Time ◽  
Convergence Of Sequences ◽  
Unique Distribution

Main goal of this paper is to formulate possibly simple and easy to verify criteria on existence of the unique attracting probability measure for stochastic process induced by generalized iterated function systems with probabilities (GIFSPs). To do this, we study the long-time behavior of trajectories of Markov-type operators acting on product of spaces of Borel measures on arbitrary Polish space. Precisely, we get the desired geometric rate of convergence of sequences of measures under the action of such operator to the unique distribution in the Hutchinson–Wasserstein distance. We apply the obtained results to study limiting behavior of random trajectories of GIFSPs as well as stochastic difference equations with multiple delays.

Sign in / Sign up

Export Citation Format

Share Document