Non-Lebesgue measurability of finite unions of Vitali selectors related to different groups

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Venuste Nyagahakwa ◽  
Gratien Haguma
Keyword(s):  
Topological Group ◽  
Finite Union ◽  
Affine Transformations ◽  
Real Numbers ◽  
Group Isomorphism ◽  
Lebesgue Measurability ◽  
Dense Subgroups

In this paper, we prove that each topological group isomorphism of the additive topological group $(\mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $\mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(\mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $\mathbb{R}$ onto itself.

scholarly journals Discrete temperature values in the sintering process as a BaTiO3-ceramics properties parameter

2017 ◽  
Vol 49 (4) ◽  
pp. 469-477
Author(s):  
Z.B. Vosika ◽  
V.V. Mitic ◽  
G.M. Lazovic ◽  
Lj. Kocic
Keyword(s):  
Time Scale ◽  
Finite Union ◽  
Sintering Process ◽  
Temperature Step ◽  
Real Numbers ◽  
Closed Intervals ◽  
Complex Process ◽  
Crystal Growth Kinetics ◽  
Basic Temperature ◽  
Kinetics Of

In this paper, we develop the new physical-mathematical time scale approach-model applied to BaTiO3-ceramics. At the beginning, a time scale is defined to be an arbitrary closed subset of the real numbers R, with the standard inherited topology. The time scale mathematical examples include real numbers R, natural numbers N, integers Z, the Cantor set (i.e. fractals), and any finite union of closed intervals of R. Calculus on time scales (TSC) was established in 1988 by Stefan Hilger. TSC, by construction, is used to describe the complex process. This method may be utilized for a description of physical, material (crystal growth kinetics, physical chemistry kinetics - for example, kinetics of barium-titanate synthesis), bio-chemical or similar systems and represents a major challenge for nowadays contemporary scientists. Generally speaking, such processes may be described by a discrete time scale. Reasonably it could be assumed that such a ?scenario? is possible for discrete temperature values as a consolidation parameter which is the basic ceramics description properties. In this work, BaTiO3-ceramics discrete temperature as thermodynamics parameter with temperature step h and the basic temperature point a is investigated. Instead of derivations, it is used backward differences with respect to temperature. The main conclusion is made towards ceramics materials temperature as description parameter.

scholarly journals Cofinitely and co-countably projective spaces

2002 ◽  
Vol 3 (2) ◽  
pp. 185
Author(s):  
Pablo Mendoza Iturralde ◽  
Vladimir V. Tkachuk
Keyword(s):  
Topological Group ◽  
Projective Space ◽  
Infinite Family ◽  
Projective Spaces ◽  
Finite Union ◽  
Paracompact Space ◽  
Finite Set ◽  
Discrete Spaces

<p>We show that X is cofinitely projective if and only if it is a finite union of Alexandroff compactatifications of discrete spaces. We also prove that X is co-countably projective if and only if X admits no disjoint infinite family of uncountable cozero sets. It is shown that a paracompact space X is co-countably projective if and only if there exists a finite set B C X such that B C U ϵ τ (X) implies │X\U│ ≤ ω. In case of existence of such a B we will say that X is concentrated around B. We prove that there exists a space Y which is co-countably projective while there is no finite set B C Y around which Y is concentrated. We show that any metrizable co-countably projective space is countable. An important corollary is that every co-countably projective topological group is countable.</p>

scholarly journals Generators for locally compact groups

1993 ◽  
Vol 36 (3) ◽  
pp. 463-467 ◽  
Author(s):  
Joan Cleary ◽  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Positive Integer ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Numbers ◽  
Locally Compact ◽  
Hausdorff Group

It is proved that if G is any compact connected Hausdorff group with weight w(G)≦c, ℝ is the topological group of all real numbers and n is a positive integer, then the topological group G × ℝn can be topologically generated by n + 1 elements, and no fewer elements will suffice.

On The Generalized Cauchy Equation

1967 ◽  
Vol 19 ◽  
pp. 1314-1318 ◽  
Author(s):  
Itrel Monroe
Keyword(s):  
Topological Group ◽  
Continuous Function ◽  
Cauchy Equation ◽  
Real Numbers ◽  
Locally Compact ◽  
The Real

It is the purpose of this note to prove the following theorem: Let ƒ: G → R be a non-constant continuous function with G a locally compact connected topological group and with R the real numbers. Let C = ƒ(G) and suppose that F: C × C → C is a junction such thatThen ƒ is monotone and open and F is continuous.

scholarly journals Some classes of minimally almost periodic topological groups

2015 ◽  
Vol 16 (2) ◽  
pp. 141 ◽  
Author(s):  
Wistar Comfort ◽  
Franklin R. Gould
Keyword(s):  
Topological Group ◽  
Almost Periodic ◽  
Topological Groups ◽  
Small Subgroup ◽  
Periodic Groups ◽  
Dense Subgroups

<p>A Hausdorff topological group G=(G,T) has the small subgroup generating property<br />(briefly: has the SSGP property, or is an SSGP group) if for each neighborhood U of $1_G$ there is a family $\sH$<br />of subgroups of $G$ such that $\bigcup\sH\subseteq U$ and<br />$\langle\bigcup\sH\rangle$ is dense in $G$. The class of $\rm{SSGP}$ groups is defined and investigated with respect<br />to the properties usually studied by topologists (products,<br />quotients, passage to dense subgroups, and the like), and with respect to the familiar class of minimally almost<br />periodic groups (the m.a.p. groups). Additional classes<br />SSGP(n) for $n&lt;\omega$ (with SSGP(1) = SSGP) are defined and investigated, and the class-theoretic inclusions<br />$$\mathrm{SSGP}(n)\subseteq\mathrm{SSGP}(n+1)\subseteq\mathrm{ m.a.p.}$$<br />are established and shown proper.</p><p>In passing the authors also establish the presence of {\rm SSGP}$(1)$ or {\rm SSGP}$(2)$ in many of the early examples in the literature of abelian m.a.p. groups.</p>

On critical pairs of product sets in a certain matrix group

1970 ◽  
Vol 67 (3) ◽  
pp. 569-581
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Finite Measure ◽  
Matrix Group ◽  
Real Numbers ◽  
Locally Compact ◽  
Image Position ◽  
Set Function ◽  
Product Sets ◽  
Critical Pairs

1. Introduction. If G is a locally compact Hausdorff topological group, and μ is (left) Haar measure on G, then we denote by ℬ(G) the class of all Borel subsets of G having finite measure, and by VG the set {μ(E): E ∊ ℬ(G)} of real numbers. The product set function of G, ΦG: VG × VG → VG, is defined (see (4) and (5)) byand, for each u, v ∈ VG, we call a pair (E, F) of Borel subsets of G a critical (u, v)-pair, if μ(E) = u, μ(F) = v, and μ*(EF) = ΦG(u, v). We denote the class of all critical (u, v)-pairs by and we write ℰG for .

scholarly journals Cardinalities of locally compact groups and their Stone-Čech compactifications

2003 ◽  
Vol 67 (3) ◽  
pp. 353-364
Author(s):  
Gerald L. Itzkowitz ◽  
Sidney A. Morris ◽  
Vladimir V. Tkachuk
Keyword(s):  
Topological Group ◽  
Discrete Group ◽  
Additive Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Numbers ◽  
Locally Compact ◽  
Euclidean Topology ◽  
Lindelöf Degree

Dedicated to Edwin HewittIf G is any Hausdorff topological group and βG is the Stone-Čech compactification then where |G| denotes the cardinalty of G It is known that if G is a discrete group then and if G is the additive group of real numbers with the Euclidean topology, then |βG| = 2|G|. In this paper the cardinality and weight of βG, for a locally compact group G, is calculated in terms of the character and Lindelöf degree of G The results make it possible to give a reasonably complete description of locally compact groups G for which |βG| = 2|G| or even |βG| = |G|.

scholarly journals A characterization of the topological group of real numbers

1986 ◽  
Vol 34 (3) ◽  
pp. 473-475 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Real Numbers ◽  
Locally Compact ◽  
Circle Group ◽  
Hausdorff Group

It is shown that a non-discrete locally compact Hausdorff group has each of its proper closed subgroups finite (respectively, discrete) if and only if it is topologically isomorphic to the circle group (respectively, the circle group or the group of real numbers).

On the Brunn-Minkowski coefficient of a locally compact unimodular group

1969 ◽  
Vol 65 (1) ◽  
pp. 33-45 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Outer Measure ◽  
Real Numbers ◽  
Locally Compact ◽  
Unimodular Group ◽  
Compact Topological Group ◽  
Image Position ◽  
Locally Compact Topological Group

Let G be a locally compact topological group, and let μ be the left Haar measure on G, with μ the corresponding outer measure. If R' denotes the non-negative extended real numbers, B (G) the Borel subsets of G, and V = {μ(C):C ∈ B(G)}, then we can define ΦG: V × V → R' bywhere AB denotes the product set of A and B in G. Then clearlyso that a knowledge of ΦG will give us some idea of how the outer measure of the product set AB compares with the measures of the sets A and B.

On product sets in a unimodular group

1968 ◽  
Vol 64 (4) ◽  
pp. 1001-1007 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Finite Measure ◽  
Real Numbers ◽  
Locally Compact ◽  
The Real ◽  
Unimodular Group ◽  
Image Position ◽  
Product Sets

Let G be a locally compact Hausdorff topological group, with µ the left Haar measure on G, and µ* the corresponding inner measure. If R denotes the real numbers, ℬ(G) denotes the Borel† subsets of G of finite measure, and VG = {µ(E):E∈ℬ(G)}, then, following Macbeath(4), we define ΦG: VG × VG→ R bywhere AB denotes the product set of A and B in G.

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