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Ignacio S Gomez - One of the best experts on this subject based on the ideXlab platform.
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a generalized Vitali Set from nonextensive statistics
Reports on Mathematical Physics, 2019Co-Authors: Ignacio S GomezAbstract:We address a generalization of the Vitali Set through a deformed translational property that stems from a generalized algebra derived from the nonextensive statistics. The generalization is based on the so-called q-addition x ⊕q y = x + y + (1 – q)xy for rational values of q, where the ordinary formalism is recovered when the control parameter q → 1. The generalized Vitali Set is non-measurable for all rational parameters 1 2 q ≤ 1 , but in the limit q → 1 2 the non-measurability cannot be guaranteed. Furthermore, assuming measurability when q → 1 2 , then this must be positive. Monotonicity, σ-additivity, σ-finiteness, and translational invariance are generalized according to the structure of the q-addition and of the q-integral.
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A generalized Vitali Set from nonextensive statistics
Reports on Mathematical Physics, 2019Co-Authors: Ignacio S GomezAbstract:We address a generalization of the Vitali Set through a deformed translational property that stems from a generalized algebra derived from the nonextensive statistics. The generalization is based on the so-called $q$-addition $x\oplus_{q} y=x+y+(1-q)xy$ for rational values of $q$, where the ordinary formalism is recovered when the control parameter $q \to 1$. The generalized Vitali Set is non-measurable for all rational parameter $\frac{1}{2}
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a generalized Vitali Set from nonextensive statistics
arXiv: Mathematical Physics, 2018Co-Authors: Ignacio S GomezAbstract:We address a generalization of the Vitali Set through a deformed translational property that stems from a generalized algebra derived from the nonextensive statistics. The generalization is based on the so-called $q$-addition $x\oplus_{q} y=x+y+(1-q)xy$ for rational values of $q$, where the ordinary formalism is recovered when the control parameter $q \to 1$. The generalized Vitali Set is non-measurable for all rational parameter $\frac{1}{2}
Terry Soo - One of the best experts on this subject based on the ideXlab platform.
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A Nonmeasurable Set from Coin Flips
American Mathematical Monthly, 2009Co-Authors: Alexander E. Holroyd, Terry SooAbstract:To motivate the elaborate machinery of measure theory, it is desirable to show that in some natural space Q one cannot define a measure on all subSets of £2, if the measure is to satisfy certain natural properties. The usual example is given by the Vitali Set, obtained by choosing one representative from each equivalence class of R induced by the relation x ~ y if and only ifx-yeQ. The resulting Set is not measurable with respect to any translation-invariant measure on R that gives nonzero, finite measure to the unit interval [8]. In particular, the resulting Set is not Lebesgue measurable. The construction above uses the axiom of choice. Indeed, the Solovay theorem [7] states that in the absence of the axiom of choice, there is a model of Zermelo-Frankel Set theory where all the subSets of R are Lebesgue measurable. In this note we give a variant proof of the existence of a nonmeasurable Set (in a slightly different space). We will use the axiom of choice in the guise of the wellordering principle (see the later discussion for more information). Other examples of nonmeasurable Sets may be found for example in [1] and [5, Ch. 5]. We will produce a nonmeasurable Set in the space Q := {0, 1}Z. Translationinvariance plays a key role in the Vitali proof. Here shift-invariance will play a similar role. The shift T : Z -> Z on integers is defined via Tx := x + 1, and the shift r : Q -> Q on elements co e Q is defined via (xco)(x) := co(x 1). We write xA := {xco : co e A] for A c Q.
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A NONMEASURABLE Set FROM COIN FLIPS
2009Co-Authors: Alexander E. Holroyd, Terry SooAbstract:To motivate the elaborate machinery of measure theory, it is desirable to show that in some natural space Ω one cannot define a measure on all subSets of Ω, if the measure is to satisfy certain natural properties. The usual example is given by the Vitali Set, obtained by choosing one representative from each equivalence class of R induced by the relation x ∼ y if and only if x −y ∈ Q. The resulting Set is not measurable with respect to any translation-invariant measure on R that gives nonzero, finite measure to the unit interval [8]. In particular, the resulting Set is not Lebesgue measurable. The construction above uses the axiom of choice. Indeed, the Solovay theorem [7] states that in the absence of the axiom of choice, there is a model of Zermelo-Frankel Set theory where all the subSets of R are Lebesgue measurable. In this note we give a variant proof of the existence of a nonmeasurable Set (in a slightly different space). We will use the axiom of choice in the guise of the well-ordering principle (see the later discussion for more information). Other examples of nonmeasurable Sets may be found for example in [1] and [5, Ch. 5]. We will produce a nonmeasurable Set in the space Ω: = {0, 1} Z. Translation-invariance plays a key role in the Vitali proof. Here shiftinvariance will play a similar role. The shift T: Z → Z on integers is defined via Tx: = x + 1, and the shift τ: Ω → Ω on elements ω ∈ Ω is defined via (τω)(x): = ω(x − 1). We write τA: = {τω: ω ∈ A} for A ⊆ Ω. Theorem 1. Let F be a σ-algebra on Ω that contains all singletons and is closed under the shift (that is, A ∈ F implies τA ∈ F). If there exists a measure µ on F that is shift-invariant (that is, µ = µ ◦ τ) and satisfies µ(Ω) ∈ (0, ∞), and µ({ω}) = 0 for all ω ∈ Ω, then F does not contain all subSets of Ω. The conditions on F and µ in Theorem 1 are indeed satisfied by measures that arise naturally. A central example is the probability space (Ω, G, P) for a sequence of independent fair coin flips indexed by Z, which is defined as follows. Let A be the algebra of all Sets of the form {ω ∈ Ω: ω(k) = ak, for all k ∈ K}, where K ⊂ Z is any finite subSet of the integers and a ∈ {0, 1} K is any finite binary string. The measur
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A NON-MEASURABLE Set FROM COIN-FLIPS
2006Co-Authors: Alexander E. Holroyd, Terry SooAbstract:To motivate the elaborate machinery of measure theory, it is desirable to have an example of a Set which is not measurable in some natural space. The usual example is the Vitali Set, obtained by picking one representative from each equivalence class of R induced by the relation x ∼ y iff x − y ∈ Q. The translation-invariance of Lebesgue measure implies that the resulting Set is not Lebesgue-measurable [4]. By the Solovay Theorem [3], one cannot construct such a Set in Zermelo-Frankel Set theory without appealing to the axiom of choice. In this note we give a variant construction in the language of probability theory, using the axiom of choice in the guise of the well-ordering principle [5]. For other constructions see [2, Ch. 5]. Consider the measure space (Ω, F, P), where Ω = {0, 1} Z, and F is the product σ-algebra, and P is the product measure (
Alexander E. Holroyd - One of the best experts on this subject based on the ideXlab platform.
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A Nonmeasurable Set from Coin Flips
American Mathematical Monthly, 2009Co-Authors: Alexander E. Holroyd, Terry SooAbstract:To motivate the elaborate machinery of measure theory, it is desirable to show that in some natural space Q one cannot define a measure on all subSets of £2, if the measure is to satisfy certain natural properties. The usual example is given by the Vitali Set, obtained by choosing one representative from each equivalence class of R induced by the relation x ~ y if and only ifx-yeQ. The resulting Set is not measurable with respect to any translation-invariant measure on R that gives nonzero, finite measure to the unit interval [8]. In particular, the resulting Set is not Lebesgue measurable. The construction above uses the axiom of choice. Indeed, the Solovay theorem [7] states that in the absence of the axiom of choice, there is a model of Zermelo-Frankel Set theory where all the subSets of R are Lebesgue measurable. In this note we give a variant proof of the existence of a nonmeasurable Set (in a slightly different space). We will use the axiom of choice in the guise of the wellordering principle (see the later discussion for more information). Other examples of nonmeasurable Sets may be found for example in [1] and [5, Ch. 5]. We will produce a nonmeasurable Set in the space Q := {0, 1}Z. Translationinvariance plays a key role in the Vitali proof. Here shift-invariance will play a similar role. The shift T : Z -> Z on integers is defined via Tx := x + 1, and the shift r : Q -> Q on elements co e Q is defined via (xco)(x) := co(x 1). We write xA := {xco : co e A] for A c Q.
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A NONMEASURABLE Set FROM COIN FLIPS
2009Co-Authors: Alexander E. Holroyd, Terry SooAbstract:To motivate the elaborate machinery of measure theory, it is desirable to show that in some natural space Ω one cannot define a measure on all subSets of Ω, if the measure is to satisfy certain natural properties. The usual example is given by the Vitali Set, obtained by choosing one representative from each equivalence class of R induced by the relation x ∼ y if and only if x −y ∈ Q. The resulting Set is not measurable with respect to any translation-invariant measure on R that gives nonzero, finite measure to the unit interval [8]. In particular, the resulting Set is not Lebesgue measurable. The construction above uses the axiom of choice. Indeed, the Solovay theorem [7] states that in the absence of the axiom of choice, there is a model of Zermelo-Frankel Set theory where all the subSets of R are Lebesgue measurable. In this note we give a variant proof of the existence of a nonmeasurable Set (in a slightly different space). We will use the axiom of choice in the guise of the well-ordering principle (see the later discussion for more information). Other examples of nonmeasurable Sets may be found for example in [1] and [5, Ch. 5]. We will produce a nonmeasurable Set in the space Ω: = {0, 1} Z. Translation-invariance plays a key role in the Vitali proof. Here shiftinvariance will play a similar role. The shift T: Z → Z on integers is defined via Tx: = x + 1, and the shift τ: Ω → Ω on elements ω ∈ Ω is defined via (τω)(x): = ω(x − 1). We write τA: = {τω: ω ∈ A} for A ⊆ Ω. Theorem 1. Let F be a σ-algebra on Ω that contains all singletons and is closed under the shift (that is, A ∈ F implies τA ∈ F). If there exists a measure µ on F that is shift-invariant (that is, µ = µ ◦ τ) and satisfies µ(Ω) ∈ (0, ∞), and µ({ω}) = 0 for all ω ∈ Ω, then F does not contain all subSets of Ω. The conditions on F and µ in Theorem 1 are indeed satisfied by measures that arise naturally. A central example is the probability space (Ω, G, P) for a sequence of independent fair coin flips indexed by Z, which is defined as follows. Let A be the algebra of all Sets of the form {ω ∈ Ω: ω(k) = ak, for all k ∈ K}, where K ⊂ Z is any finite subSet of the integers and a ∈ {0, 1} K is any finite binary string. The measur
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A NON-MEASURABLE Set FROM COIN-FLIPS
2006Co-Authors: Alexander E. Holroyd, Terry SooAbstract:To motivate the elaborate machinery of measure theory, it is desirable to have an example of a Set which is not measurable in some natural space. The usual example is the Vitali Set, obtained by picking one representative from each equivalence class of R induced by the relation x ∼ y iff x − y ∈ Q. The translation-invariance of Lebesgue measure implies that the resulting Set is not Lebesgue-measurable [4]. By the Solovay Theorem [3], one cannot construct such a Set in Zermelo-Frankel Set theory without appealing to the axiom of choice. In this note we give a variant construction in the language of probability theory, using the axiom of choice in the guise of the well-ordering principle [5]. For other constructions see [2, Ch. 5]. Consider the measure space (Ω, F, P), where Ω = {0, 1} Z, and F is the product σ-algebra, and P is the product measure (
Daniel R. Mauldin - One of the best experts on this subject based on the ideXlab platform.
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On the unique representation of families of Sets
2015Co-Authors: Su Gao, Steve Jackson, Daniel R. MauldinAbstract:Abstract. Let X and Y be uncountable Polish spaces. A ⊂ X×Y represents a family of Sets C provided each Set in C occurs as an x-section of A. We say A uniquely represents C provided each Set in C occurs exactly once as an x-section of A. A is universal for C if every x-section of A is in C. A is uniquely universal for C if it is universal and uniquely represents C. We show that there is a Borel Set in X × R which uniquely represents the translates of Q if and only if there is a Σ12 Vitali Set. Assuming V = L there is a Borel Set B ⊂ ωω with all sections Fσ Sets and all non-empty Kσ Sets are uniquely represented by B. Assuming V = L there is a Borel Set B ⊂ X × Y with all sections Kσ which uniquely represents the countable subSets of Y. There is an analytic Set in X×Y with all sections ∆02 which represents all the ∆02 subSets of Y, but no Borel Set can uniquely represent the ∆02 Sets. This last thoerem is generalized to higher Borel classes. 1
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ON THE UNIQUE REPRESENTATION OF FAMILIES OF SetS
2008Co-Authors: Su Gao, Steve Jackson, Miklos Laczkovich, Daniel R. MauldinAbstract:Abstract. Let X and Y be uncountable Polish spaces. A ⊂ X ×Y represents a family of Sets C provided each Set in C occurs as an x-section of A. We say A uniquely represents C provided each Set in C occurs exactly once as an xsection of A. A is universal for C if A represents C and every x-section of A is in C. A is uniquely universal for C if it is universal and uniquely represents C. We show that there is a Borel Set in X × R which uniquely represents the translates of Q if and only if there is a Σ1 2 Vitali Set. Assuming V = L there is a Borel Set B ⊂ ωω with all sections Fσ Sets and all non-empty Kσ Sets are uniquely represented by B. Assuming V = L there is a Borel Set B ⊂ X × Y with all sections Kσ which uniquely represents the countable subSets of Y. There is an analytic Set in X × Y with all sections ∆0 2 which represents all the ∆0 2 subSets of Y, but no Borel Set can uniquely represent the ∆02 Sets. This last theorem is generalized to higher Borel classes. 1
Steve Jackson - One of the best experts on this subject based on the ideXlab platform.
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On the unique representation of families of Sets
2015Co-Authors: Su Gao, Steve Jackson, Daniel R. MauldinAbstract:Abstract. Let X and Y be uncountable Polish spaces. A ⊂ X×Y represents a family of Sets C provided each Set in C occurs as an x-section of A. We say A uniquely represents C provided each Set in C occurs exactly once as an x-section of A. A is universal for C if every x-section of A is in C. A is uniquely universal for C if it is universal and uniquely represents C. We show that there is a Borel Set in X × R which uniquely represents the translates of Q if and only if there is a Σ12 Vitali Set. Assuming V = L there is a Borel Set B ⊂ ωω with all sections Fσ Sets and all non-empty Kσ Sets are uniquely represented by B. Assuming V = L there is a Borel Set B ⊂ X × Y with all sections Kσ which uniquely represents the countable subSets of Y. There is an analytic Set in X×Y with all sections ∆02 which represents all the ∆02 subSets of Y, but no Borel Set can uniquely represent the ∆02 Sets. This last thoerem is generalized to higher Borel classes. 1
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ON THE UNIQUE REPRESENTATION OF FAMILIES OF SetS
2008Co-Authors: Su Gao, Steve Jackson, Miklos Laczkovich, Daniel R. MauldinAbstract:Abstract. Let X and Y be uncountable Polish spaces. A ⊂ X ×Y represents a family of Sets C provided each Set in C occurs as an x-section of A. We say A uniquely represents C provided each Set in C occurs exactly once as an xsection of A. A is universal for C if A represents C and every x-section of A is in C. A is uniquely universal for C if it is universal and uniquely represents C. We show that there is a Borel Set in X × R which uniquely represents the translates of Q if and only if there is a Σ1 2 Vitali Set. Assuming V = L there is a Borel Set B ⊂ ωω with all sections Fσ Sets and all non-empty Kσ Sets are uniquely represented by B. Assuming V = L there is a Borel Set B ⊂ X × Y with all sections Kσ which uniquely represents the countable subSets of Y. There is an analytic Set in X × Y with all sections ∆0 2 which represents all the ∆0 2 subSets of Y, but no Borel Set can uniquely represent the ∆02 Sets. This last theorem is generalized to higher Borel classes. 1
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on the unique representation of families of Sets
Transactions of the American Mathematical Society, 2008Co-Authors: Steve Jackson, Miklos Laczkovich, R MauldinAbstract:Let X and Y be uncountable Polish spaces. A. X x Y represents a family of Sets C provided each Set in C occurs as an x-section of A. We say that A uniquely represents C provided each Set in C occurs exactly once as an x-section of A. A is universal for C if every x-section of A is in C. A is uniquely universal for C if it is universal and uniquely represents C. We show that there is a Borel Set in X x R which uniquely represents the translates of Q if and only if there is a Sigma(1)(2) Vitali Set. Assuming V = L there is a Borel Set B subSet of omega(omega). with all sections F sigma Sets and all non-empty K sigma Sets are uniquely represented by B. Assuming V = L there is a Borel Set B subSet of X x Y with all sections Ks which uniquely represents the countable subSets of Y. There is an analytic Set in X x Y with all sections Delta(0)(2) which represents all the Delta(0)(2) subSets of Y, but no Borel Set can uniquely represent the Delta(0)(2) Sets. This last theorem is generalized to higher Borel classes.
