Visualizing a Nonmeasurable Set

2003 ◽  
Vol 76 (2) ◽  
pp. 137
Author(s):  
Saleem Watson ◽  
Arthur Wayman
Keyword(s):  
Nonmeasurable Set

scholarly journals On Negligible and Absolutely Nonmeasurable Subsets of the Euclidean Plane

2004 ◽  
Vol 11 (3) ◽  
pp. 479-487
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Plane ◽  
Identity Transformation ◽  
Extension Problem ◽  
Measure Extension ◽  
Nonmeasurable Set

Abstract The notions of a negligible set and of an absolutely nonmeasurable set are introduced and discussed in connection with the measure extension problem. In particular, it is demonstrated that there exist subsets of the plane 𝐑2 which are 𝑇2-negligible and, simultaneously, 𝐺-absolutely nonmeasurable. Here 𝑇2 denotes the group of all translations of 𝐑2 and 𝐺 denotes the group generated by {𝑔} ∪ 𝑇2, where 𝑔 is an arbitrary rotation of 𝐑2 distinct from the identity transformation and all central symmetries of 𝐑2.

Visualizing a Nonmeasurable Set

2003 ◽  
Vol 76 (2) ◽  
pp. 137-140
Author(s):  
Saleem Watson ◽  
Arthur Wayman
Keyword(s):  
Nonmeasurable Set

scholarly journals On Sierpiński's nonmeasurable set

1991 ◽  
Vol 139 (1) ◽  
pp. 17-22
Author(s):  
Marc Frantz
Keyword(s):  
Nonmeasurable Set

On Definability of Nonmeasurable Sets

1980 ◽  
Vol 32 (3) ◽  
pp. 653-656 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Equivalence Class ◽  
Inaccessible Cardinal ◽  
Generic Extension ◽  
Nonmeasurable Set ◽  
Nonmeasurable Sets

In [1], Solovay constructed a model of ZFC in which every set of reals in OD (R) is Lebesgue measurable. Here we construct a model in which every equivalence class of sets of reals modulo null sets that is in OD(R) consists of Lebesgue measurable sets. This result immediately implies Solovay's, since the equivalence class of any set of reals in OD (R) is itself OD (R). As a consequence, one cannot provably explicitly define a nonmeasurable set modulo null sets within ZFC. We do not know whether this holds in the model Solovay uses (where an inaccessible cardinal is collapsed to ω 1). Instead, our model is a generic extension of his model.In the model we construct, a somewhat stronger statement holds: every set in OD(R) of sets of reals which has < 2C inequivalent elements modulo null sets, consists entirely of Lebesgue measurable sets of reals.

On measurability of algebraic sums of small sets

2008 ◽  
Vol 45 (3) ◽  
pp. 433-442
Author(s):  
Alexander Kharazishvili ◽  
Aleks Kirtadze
Keyword(s):  
Real Line ◽  
Measure Zero ◽  
The Real ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Nonmeasurable Set ◽  
Universal Measure

It is shown that, under Martin’s Axiom, the algebraic sum of two universal measure zero subsets of the real line can be an absolutely nonmeasurable set. Some related questions concerning measurability of algebraic sums of small sets are also discussed.

ANOTHER NONMEASURABLE SET WITH PROPERTY (s⁰)

1991 ◽  
Vol 17 (2) ◽  
pp. 781 ◽  
Author(s):  
Balcerzak
Keyword(s):  
Nonmeasurable Set

On Absolutely Nonmeasurable Sets and Functions

2008 ◽  
Vol 15 (2) ◽  
pp. 317-325
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Nonmeasurable Set ◽  
Set Function ◽  
Nonmeasurable Sets

Abstract We consider some properties of sets and functions which are measurable (or nonmeasurable) with respect to certain classes of measures. In this context, the notion of an absolutely nonmeasurable set (function) is examined. Sierpiński-Zygmund type functions are constructed having additional properties closely connected with the above-mentioned notion. Also, some small subsets of uncountable commutative groups are discussed whose algebraic sum turns out to be an absolutely nonmeasurable set.

scholarly journals A Nonmeasurable Set from Coin Flips

2009 ◽  
Vol 116 (10) ◽  
pp. 926-928 ◽  
Author(s):  
Alexander E. Holroyd ◽  
Terry Soo
Keyword(s):  
Nonmeasurable Set
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