scholarly journals THE GAME OPERATOR ACTING ON WADGE CLASSES OF BOREL SETS

2019 ◽  
Vol 84 (3) ◽  
pp. 1224-1239
Author(s):  
GABRIEL DEBS ◽  
JEAN SAINT RAYMOND
Keyword(s):  
Borel Sets ◽  
Substitution Property ◽  
Image Position

AbstractWe study the behavior of the game operator $$ on Wadge classes of Borel sets. In particular we prove that the classical Moschovakis results still hold in this setting. We also characterize Wadge classes ${\bf{\Gamma }}$ for which the class has the substitution property. An effective variation of these results shows that for all $1 \le \eta < \omega _1^{{\rm{CK}}}$ and $2 \le \xi < \omega _1^{{\rm{CK}}}$, is a Spector class while is not.

Sufficient Statistics and Orthogonal Parameters

1962 ◽  
Vol 58 (2) ◽  
pp. 326-337 ◽  
Author(s):  
Ann F. S. Mitchell
Keyword(s):  
Probability Density Function ◽  
Finite Measure ◽  
Random Variable ◽  
Sample Space ◽  
Continuous Random Variable ◽  
Sufficient Statistics ◽  
Counting Measure ◽  
Borel Sets ◽  
Orthogonal Parameters ◽  
Image Position

Let be, for a set of n real continuous parameters the probability density function of a random variable x with respect to a σ-finite measure μ on a σ-algebra of subsets of the sample space . If x; is a continuous random variable, μ will be Lebesgue measure on the Borel sets of a Euclidean sample space and, if x is discrete, μ will be counting measure on the class of all sets of a countable sample space. The parameters αi are said to be orthogonal (Jeffreys (3), pp. 158,184) if .

Ergodic Averages for Weight Functions Moved by Non-Linear Transformations on ℝn

1995 ◽  
Vol 47 (4) ◽  
pp. 852-876
Author(s):  
David I. McIntosh
Keyword(s):  
Sufficient Conditions ◽  
Convergent Sequence ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Linear Transformations ◽  
Borel Sets ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Sets

AbstractLet ℝ+ denote the non-negative half of the real line, and let λ denote Lebesgue measure on the Borel sets of ℝn. A function φ: ℝn → ℝ+ is called a weight function if ʃℝn φ dλ = 1. Let (X, ℱ, μ) be a non-atomic, finite measure space, let ƒ: X → ℝ+, and suppose { Tν}ν∊ℝn is an ergodic, aperiodic ℝn-flow on X. We consider the weighted ergodic averages where is a sequence of weight functions. Sufficient as well as necessary and sufficient conditions for the pointwise, almost-everywhere convergence of are developed for a particular class of weight functions φk. Specifically, let {τk: ℝn → ℝn} be a sequence of measurable, non-singular maps with measurable, non-singular inverses such that the Radon-Nikodym derivatives dλ oτk /dλ and dλ oτk-1 / dλ are L∞ (ℝn), and such that τk and τ-1 map bounded sets to bounded sets. We examine convergence for the sequence where θk is an a.e.-convergent sequence of weight functions which are dominated by a fixed L1(ℝn) function with bounded support.

7.—Convolution of Vector Measures

1975 ◽  
Vol 73 ◽  
pp. 117-135 ◽  
Author(s):  
A. J. White
Keyword(s):  
Banach Algebra ◽  
Large Class ◽  
Independent Interest ◽  
Scalar Case ◽  
Algebra Structure ◽  
Semi Group ◽  
Locally Compact ◽  
Vector Measures ◽  
Borel Sets ◽  
Image Position

SynopsisIn this paper we consider the problem of defining a convolution, analogous to the classical convolution of scalar measures, of measures defined on the Borel sets of a locally compact semi-groupSand having values in a Banach algebra. Using the bilinearintegral introduced by Bartle we show that the formalism of the scalar case persists in situations of considerable generality so that the formulasuitably interpreted, gives a Banach algebra structure to a large class ofvalued measures defined onS.Themethods exploit the connection between vector measures and operators and involve some results of independent interest.

Borel sets and Ramsey's theorem

1973 ◽  
Vol 38 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Fred Galvin ◽  
Karel Prikry
Keyword(s):  
Recursion Theory ◽  
Main Result Theorem ◽  
Borel Sets ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Power Set ◽  
Definable Sets ◽  
Result Theorem ◽  
Image Position ◽  
Usual Product

Definition 1. For a set S and a cardinal κ,In particular, 2ω denotes the power set of the natural numbers and not the cardinal 2ℵ0. We regard 2ω as a topological space with the usual product topology.Definition 2. A set S ⊆ 2ω is Ramsey if there is an M ∈ [ω]ω such that either [M]ω ⊆ S or else [M]ω ⊆ 2ω − S.Erdös and Rado [3, Example 1, p. 434] showed that not every S ⊆ 2ω is Ramsey. In view of the nonconstructive character of the counterexample, one might expect (as Dana Scott has suggested) that all sufficiently definable sets are Ramsey. In fact, our main result (Theorem 2) is that all Borei sets are Ramsey. Soare [10] has applied this result to some problems in recursion theory.The first positive result on Scott's problem was Ramsey's theorem [8, Theorem A]. The next advance was Nash-Williams' generalization of Ramsey's theorem (Corollary 2), which can be interpreted as saying: If S1 and S2 are disjoint open subsets of 2ω, there is an M ∈ [ω]ω such that either [M]ω ⋂ S1 = ∅ or [M]ω ∩ S2 = ⊆. (This is halfway between “clopen sets are Ramsey” and “open sets are Ramsey.”) Then Galvin [4] stated a generalization of Nash-Williams' theorem (Corollary 1) which says, in effect, that open sets are Ramsey; this was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.

On diagonal values of probability vectors of infinitely many components

1961 ◽  
Vol 57 (4) ◽  
pp. 759-766
Author(s):  
A. S. Besicovitch
Keyword(s):  
Finite Number ◽  
Full Range ◽  
Dimensional Vector ◽  
Probability Vector ◽  
Real Numbers ◽  
Borel Sets ◽  
Image Position ◽  
Borel Class ◽  
Vector Valued ◽  
Range Of Values

Given † probability vector μ(X) = (μ1(X), … μk (X)) of a finite number of components on a Borel class of sets X, we say that μ(X0) has a diagonal value α if μi(X0) = α for all i = 1, 2,…,K. J. Neyman(l), (2), (3) has proved that in the class of Borel sets of real numbers any non-atomic vector μ(X) takes all diagonal values. A. Liapounoff has studied the full range of values of k-dimensional vector-valued measures and in two papers (4) he has proved that the range is closed and in the case of non-atomic measures the range is also convex. He also gave an example showing that neither of these results holds in the case of vectors of infinitely many components. A simplified proof of Liapounoff's results has been given by P. R. Halmos (5). In the present paper I study the range of values of probability vectors of infinitely many components. Various types of conditions are studied which are sufficient to imply that, for each ε > 0, 0 ≤ α ≤ 1, it is possible to find a set X such that

On semi-Markov processes on arbitrary spaces

1969 ◽  
Vol 66 (2) ◽  
pp. 381-392 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Markov Processes ◽  
Borel Sets ◽  
Image Position ◽  
Semi Markov Processes

Let E be an arbitrary set, a σ-algebra of subsets of the Borel sets of F. We write A × B for the product of the sets A and B, andfor the product σ-algebra of and (i.e. the σ-algebra generated by the rectangles A × B with A ∈ and B ∈ ).

WADGE HIERARCHY OF DIFFERENCES OF CO-ANALYTIC SETS

2016 ◽  
Vol 81 (1) ◽  
pp. 201-215 ◽  
Author(s):  
KEVIN FOURNIER
Keyword(s):  
Baire Space ◽  
Borel Sets ◽  
Wadge Hierarchy ◽  
Analytic Sets ◽  
Image Position

AbstractWe begin the fine analysis of nonBorel pointclasses. Working in ZFC + DET$\left( {_1^1 } \right)$, we describe the Wadge hierarchy of the class of increasing differences of co-analytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets.

Extending Baire property by uncountably many sets

2010 ◽  
Vol 75 (3) ◽  
pp. 896-904
Author(s):  
Paweł Kawa ◽  
Janusz Pawlikowski
Keyword(s):  
Lebesgue Measure ◽  
Baire Property ◽  
Borel Sets ◽  
Real Model ◽  
Cohen Real ◽  
Image Position ◽  
Random Reals ◽  
Meager Sets

AbstractWe show that for an uncountable κ in a suitable Cohen real model for any family {Av}v<κ of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets Av, into the algebra of Baire subsets of 2κ modulo meager sets such that for all Borel B,The proof is uniform, works also for random reals and the Lebesgue measure, and in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

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