scholarly journals Measures equivalent to the Haar measure

1960 ◽  
Vol 4 (4) ◽  
pp. 157-162 ◽  
Author(s):  
S. Świerczkowski
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
The Other ◽  
Compact Sets ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Open Set ◽  
Baire Measure ◽  
Two Measures ◽  
Locally Compact Topological Group

We call two measures equivalent if each is absolutely continuous with respect to the other (cf. [1]). Let G be a locally compact topological group and let μ be a non-negative Baire measure on G (i.e. μ is denned on all Baire sets, finite on compact sets and positive on open sets). We say that μ is stable if μ (E)=0 implies μ(tE)=0 for each t ∈ G. A. M. Macbeath made the conjecture that every stable non-trivial Baire measure is equivalent to the Haar measure. In this paper we prove the following slightly stronger result:Theorem. Every stable non-trivial measure defined on Baire sets and finite on some open set is equivalent to the Haar measure.

scholarly journals On Theorems of Kawada and Wendel

1958 ◽  
Vol 11 (2) ◽  
pp. 71-77 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Convolution Product ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Complex Functions ◽  
Image Position ◽  
Locally Compact Topological Group ◽  
Invariant Haar Measure

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li byis again in L1, and

Lp(G)-isotone measures

1975 ◽  
Vol 78 (3) ◽  
pp. 471-481 ◽  
Author(s):  
Beryl J. Peers
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Equivalence Classes ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Borel Measures ◽  
Integrable Functions ◽  
Locally Compact Topological Group

Let G be a locally compact topological group with left Haar measure, m; let M(G) denote the bounded regular Borel measures on G and let Lp(G) denote the equivalence classes of pth power integrable functions on G with respect to the left Haar measure.

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.

scholarly journals CONTINUITY OF MEASURABLE HOMOMORPHISMS

2008 ◽  
Vol 78 (1) ◽  
pp. 171-176 ◽  
Author(s):  
JANUSZ BRZDȨK
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Baire Property ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

AbstractWe give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

scholarly journals Certain semigroups embeddabable in topological groups

1982 ◽  
Vol 33 (1) ◽  
pp. 30-39 ◽  
Author(s):  
Heneri A. M. Dzinotyiweyi
Keyword(s):  
Invariant Measures ◽  
Locally Compact Group ◽  
Continuous Measure ◽  
Topological Groups ◽  
Empty Interior ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Topological Semigroups ◽  
Absolutely Continuous Measure ◽  
Locally Compact Topological Group

AbstractIn this paper we study commutative topological semigroups S admitting an absolutely continuous measure. When S is cancellative we show that S admits a weaker topology J with respect to which (S, J) is embeddable as a subsemigroup with non-empty interior in some locally compact topological group. As a consequence, we deduce certain results related to the existence of invariant measures on S and for a large class of locally compact topological semigroups S, we associate S with some useful topological subsemigroup of a locally compact group.

Locally compact groups with closed subgroups open and p-adic

1995 ◽  
Vol 118 (2) ◽  
pp. 303-313 ◽  
Author(s):  
Karl H. Hofmann ◽  
Sidney A. Morris ◽  
Sheila Oates-Williams ◽  
V. N. Obraztsov
Keyword(s):  
Topological Group ◽  
Open Subgroup ◽  
Additive Group ◽  
Locally Compact Abelian Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Character Group ◽  
Exceptional Groups ◽  
Open Set ◽  
Compact Abelian Groups

An open subgroup U of a topological group G is always closed, since U is the complement of the open set . An arbitrary closed subgroup C of G is almost never open, unless G belongs to a small family of exceptional groups. In fact, if G is a locally compact abelian group in which every non-trivial subgroup is open, then G is the additive group δp of p-adic integers or the additive group Ωp of p-adic rationale (cf. Robertson and Schreiber[5[, proposition 7). The fact that δp has interesting properties as a topological group has many roots. One is that its character group is the Prüfer group ℤp∞, which makes it unique inside the category of compact abelian groups. But even within the bigger class of not necessarily abelian compact groups the p-adic group δp is distinguished: it is the only one all of whose non-trivial subgroups are isomorphic (cf. Morris and Oates-Williams[2[), and it is also the only one all of whose non-trivial closed subgroups have finite index (cf. Morris, Oates-Williams and Thompson [3[).

scholarly journals On topological groups with remainder of character k

2016 ◽  
Vol 17 (1) ◽  
pp. 51
Author(s):  
Maddalena Bonanzinga ◽  
Maria Vittoria Cuzzupè
Keyword(s):  
Topological Group ◽  
Infinite Cardinal ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

<p style="margin: 0px;">In [A.V. Arhangel'skii and J. van Mill, On topological groups with a first-countable remainder, Top. Proc. <span id="OBJ_PREFIX_DWT1099_com_zimbra_phone" class="Object">42 (2013), 157-163</span>] it is proved that the character of a non-locally compact topological group with a first countable remainder doesn't exceed $\omega_1$ and a non-locally compact topological group of character $\omega_1$ having a compactification whose reminder is first countable is given. We generalize these results in the general case of an arbitrary infinite cardinal k.</p><p style="margin: 0px;"> </p>

Actions of a Locally Compact Group with Zero

1971 ◽  
Vol 23 (3) ◽  
pp. 413-420 ◽  
Author(s):  
T. H. McH. Hanson
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Topological Semigroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Definition Of ◽  
Isolated Point ◽  
Locally Compact Topological Group

In [2] we find the definition of a locally compact group with zero as a locally compact Hausdorff topological semigroup, S, which contains a non-isolated point, 0, such that G = S – {0} is a group. Hofmann shows in [2] that 0 is indeed a zero for S, G is a locally compact topological group, and the unit, 1, of G is the unit of S. We are to study actions of S and G on spaces, and the reader is referred to [4] for the terminology of actions.If X is a space (all are assumed Hausdorff) and A ⊂ X, A* denotes the closure of A. If {xρ} is a net in X, we say limρxρ = ∞ in X if {xρ} has no subnet which converges in X.

Compactness of a Locally Compact Group G and Geometric Properties of Ap(G)

1996 ◽  
Vol 48 (6) ◽  
pp. 1273-1285 ◽  
Author(s):  
Tianxuan Miao
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Multiplication Operator ◽  
Locally Compact Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Geometric Properties ◽  
Nikodym Property ◽  
Group A ◽  
Locally Compact Topological Group

AbstractLet G be a locally compact topological group. A number of characterizations are given of the class of compact groups in terms of the geometric properties such as Radon-Nikodym property, Dunford-Pettis property and Schur property of Ap(G), and the properties of the multiplication operator on PFp(G). We extend and improve several results of Lau and Ülger [17] to Ap(G) and Bp(G) for arbitrary p.

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