Nonmeasurable Set

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Terry Soo - One of the best experts on this subject based on the ideXlab platform.

  • A Nonmeasurable Set from Coin Flips
    American Mathematical Monthly, 2009
    Co-Authors: Alexander E. Holroyd, Terry Soo
    Abstract:

    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.

  • A Nonmeasurable Set FROM COIN FLIPS
    2009
    Co-Authors: Alexander E. Holroyd, Terry Soo
    Abstract:

    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

  • a Nonmeasurable Set from coin flips
    arXiv: Probability, 2006
    Co-Authors: Alexander E. Holroyd, Terry Soo
    Abstract:

    In this note we give an example of a Nonmeasurable Set in the probability space for an infinite sequence of coin flips. The example arises naturally from the notion of an equivariant function, and serves as a pedagogical illustration of the need for measure theory.

Alexander Kharazishvili - One of the best experts on this subject based on the ideXlab platform.

Soo Terry - One of the best experts on this subject based on the ideXlab platform.

  • Coupling, matching, and equivariance
    University of British Columbia, 2010
    Co-Authors: Soo Terry
    Abstract:

    This thesis consists of four research papers and one expository note that study factors of point processes in the contexts of thinning and matching. In "Poisson Splitting by Factors," we prove that given a Poisson point process on Rd with intensity λ, as a deterministic function of the process, we can colour the points red and blue, so that each colour class forms a Poisson point process on Rd, with any given pair of intensities summing λ; furthermore, the function can be chosen as an isometry-equivariant finitary factor (that is, if an isometry is applied to the points of the original process the points are still coloured the same way). Thus using only local information, without a central authority or additional randomization, points of a Poisson process can split into two groups, each of which are still Poisson. In "Deterministic Thinning of Finite Poisson Processes," we investigate similar questions for Poisson point processes on a finite volume. In this Setting we find that even without considerations of equivariance, thinning can not always be achieved as a deterministic function of the Poisson process and the existence of such a function depends on the intensities of the original and resulting Poisson process. In "Insertion and Deletion Tolerance of Point Processes," we define for point processes a version of the concept of finite-energy. This simple concept has many interesting consequences. We explore the consequences of having finite-energy in the contexts of the Boolean continuum percolation model, Palm theory and stable matchings of point processes. In "Translation-Equivariant Matchings of Coin-Flips on Zd," as a factor of i.i.d. fair coin-flips on Zd, we construct perfect matchings of heads and tails and prove power law upper bounds on the expected distance between matched pairs. In the expository note "A Nonmeasurable Set from Coin-Flips," using the notion of an equivariant function, we give an example of a Nonmeasurable Set in the probability space for an infinite sequence of coin-flips.Science, Faculty ofMathematics, Department ofGraduat

  • A Nonmeasurable Set from Coin Flips
    2009
    Co-Authors: Holroyd, Alexander E., Soo Terry
    Abstract:

    In this note we give an example of a Nonmeasurable Set in the probability space for an infinite sequence of coin flips. The example arises naturally from the notion of an equivariant function, and serves as a pedagogical illustration of the need for measure theory.Comment: 4 page

Alexander E. Holroyd - One of the best experts on this subject based on the ideXlab platform.

  • A Nonmeasurable Set from Coin Flips
    American Mathematical Monthly, 2009
    Co-Authors: Alexander E. Holroyd, Terry Soo
    Abstract:

    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.

  • A Nonmeasurable Set FROM COIN FLIPS
    2009
    Co-Authors: Alexander E. Holroyd, Terry Soo
    Abstract:

    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

  • a Nonmeasurable Set from coin flips
    arXiv: Probability, 2006
    Co-Authors: Alexander E. Holroyd, Terry Soo
    Abstract:

    In this note we give an example of a Nonmeasurable Set in the probability space for an infinite sequence of coin flips. The example arises naturally from the notion of an equivariant function, and serves as a pedagogical illustration of the need for measure theory.

Holroyd, Alexander E. - One of the best experts on this subject based on the ideXlab platform.

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