scholarly journals UNIFORM EXHAUSTIVITY CRITERIA OF A FAMILY OF VECTOR OUTER MEASURES

2017 ◽  
Vol 18 (6) ◽  
pp. 58-65
Author(s):  
T.A. Sribnaya
Keyword(s):  
Topological Group ◽  
Complete Class ◽  
Abelian Topological Group

The uniform exhaustivity criteria are proved for a sequence of exhaustive outer measures defined on the non-sigma-complete class of sets and taking values in an Abelian topological group.

scholarly journals Convergence of ergodic averages for many group rotations

2015 ◽  
Vol 36 (7) ◽  
pp. 2107-2120
Author(s):  
ZOLTÁN BUCZOLICH ◽  
GABRIELLA KESZTHELYI
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Measurable Function ◽  
Dual Group ◽  
Ergodic Averages ◽  
Abelian Topological Group ◽  
Compact Abelian Topological Group ◽  
Image Position ◽  
Rotation Set ◽  
Group Rotations

Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f:G\rightarrow \mathbb{R}$ is a measurable function. Given $(n_{k})$, a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages $$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$ The $f$-rotation set is $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$We prove that if $G$ is a compact locally connected Abelian group and $f:G\rightarrow \mathbb{R}$ is a measurable function then from $m(\unicode[STIX]{x1D6E4}_{f})>0$ it follows that $f\in L^{1}(G)$. A similar result is established for ordinary Birkhoff averages if $G=Z_{p}$, the group of $p$-adic integers. However, if the dual group, $\widehat{G}$, contains ‘infinitely many multiple torsion’ then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, $f(x+n_{k}\unicode[STIX]{x1D6FC})/k$, $k=1,\ldots ,$ for almost every $x$ for many $\unicode[STIX]{x1D6FC}$; hence, some of our theorems are stated by using instead of $\unicode[STIX]{x1D6E4}_{f}$ slightly larger sets, denoted by $\unicode[STIX]{x1D6E4}_{f,b}$.

scholarly journals VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

2008 ◽  
Vol 78 (3) ◽  
pp. 487-495 ◽  
Author(s):  
CAROLYN E. MCPHAIL ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Scattered Spaces

AbstractThe variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

Open subgroups of free abelian topological groups

1993 ◽  
Vol 114 (3) ◽  
pp. 439-442 ◽  
Author(s):  
Sidney A. Morris ◽  
Vladimir G. Pestov
Keyword(s):  
Topological Group ◽  
Sufficient Conditions ◽  
Open Subgroup ◽  
Regular Space ◽  
Topological Groups ◽  
Covering Dimension ◽  
Abelian Topological Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Free Abelian Topological Group

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

EMBEDDING OF THE FREE ABELIAN TOPOLOGICAL GROUP INTO

Mathematika ◽  
2019 ◽  
Vol 65 (3) ◽  
pp. 708-718 ◽  
Author(s):  
Mikołaj Krupski ◽  
Arkady Leiderman ◽  
Sidney Morris
Keyword(s):  
Topological Group ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group

SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

2017 ◽  
Vol 97 (1) ◽  
pp. 110-118 ◽  
Author(s):  
SAAK S. GABRIYELYAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Vector Space ◽  
Topological Vector Space ◽  
Unit Interval ◽  
Vector Subspace ◽  
Locally Compact ◽  
Tychonoff Space ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Usual Topology

For a Tychonoff space $X$, let $\mathbb{V}(X)$ be the free topological vector space over $X$, $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$, then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

scholarly journals Groups – Additive Notation

2015 ◽  
Vol 23 (2) ◽  
pp. 127-160 ◽  
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Group ◽  
Group Theory ◽  
Abelian Group ◽  
Normal Subgroup ◽  
Dense Subset ◽  
Abelian Topological Group ◽  
Covering Group ◽  
Mizar Mathematical Library ◽  
Order Of An Element ◽  
Generic Theory

Abstract We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.

Free subgroups of free abelian topological groups

1986 ◽  
Vol 100 (2) ◽  
pp. 347-353 ◽  
Author(s):  
E. Katz ◽  
S. A. Morris ◽  
P. Nickolas
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Free Subgroups ◽  
General Conditions

In this paper we prove a theorem which gives general conditions under which the free abelian topological group F(Y) on a space Y can be embedded in the free abeian topological group F(X) on a space X.

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