Triangles and Vitali Sets

We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set

We prove that there exists a special homeomorphism of the Cantor space such that every noncancellable composition of finite powers and translations of rational numbers has no fixed point. For this homeomorphism there exists both a Vitali and Bernstein subset of the Cantor set such that the image of this set is equal to its complement. There exists a Bernstein and Vitali set such that there is no Borel isomorphism between this set and its complement.

It is well known that any Vitali set on the real line ℝ does not possess the Baire property. The same is valid for finite unions of Vitali sets. What can be said about infinite unions of Vitali sets? Let S be a Vitali set, Sr be the image of S under the translation of ℝ by a rational number r and F = {Sr: r is rational}. We prove that for each non-empty proper subfamily F′ of F the union ∪F′ does not possess the Baire property. We say that a subset A of ℝ possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ⊃ O \ M. Then we characterize those non-empty proper subfamilies F′ of F which unions ∪F′ possess the Vitali property.

Let A be the family of all meager sets of the real line ℝ, V be the family of all Vitali sets of ℝ, V1be the family of all finite unions of elements of V and V2 = {(C \ A1) ∪ A2: C ∈ V1; A1, A2 ∈ A ...

If [Formula: see text] is a closed Noetherian graph on a [Formula: see text]-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF[Formula: see text][Formula: see text][Formula: see text]DC that [Formula: see text] is countably chromatic and there is no Vitali set.

The paper deals with the measurability properties of some classical subsets of the real line ℝ having an extra-ordinary descriptive structure: Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpiński sets. In particular, it is shown that there exists a translation invariant measure μ on ℝ extending the Lebesgue measure and such that all Sierpiński sets are measurable with respect to μ.

It is shown that some Vitali subsets of the real line R can be measurable with respect to certain translation quasi-invariant measures on R extending the standard Lebesgue measure. On the other hand, there exist Vitali sets which are nonmeasurable with respect to every nonzero σ-finite translation quasi-invariant measure on R.

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 Generalized Vitali Set from Nonextensive Statistics

It is shown that the cardinality of the continuum is not real-valued measurable if and only if there exists no nonzero σ-finite diffused measure μ on the real line such that all Vitali sets (respectively all Bernstein sets) are μ-measurable.

Erratum to: “On the families of sets without the Baire property generated by the vitali sets”

Let R denote the set of real numbers. A Vitali subset of R is defined to be a set of representatives of the equivalence classes given by the relation ∼ on R defined as a∼b if and only if a – b is a...

We extend Vitali’s procedure to get nonmeasurable sets. Answering a question of Kharazishvili [4], we present a relatively simple argument for passing from singletons to finite sets. Our result is actually much stronger, as it covers even the case of compact choices.

In the previous chapters we were concerned with various extensions of ??-finite measures, which are obtained by using certain types of nonmeasurable sets or nonmeasurable functions. It was also mentioned that sometimes absolutely nonmeasurable sets can occur for concrete classes of measures (e.g., Vitali sets for the class of all translation-invariant extensions of the Lebesgue measure λ=λ1 on the real line R = R1). Obviously, the latter sets turn out to be useless from the point of view of the measure extension problem.KeywordsConvex BodyReal AnalysisSeparable Banach SpaceCompact Topological GroupCountable SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A Note on Generalized Vitali Sets with Respect to Some Arbitrary Deformed Sums