scholarly journals An application of a theorem of Singer

1974 ◽  
Vol 19 (2) ◽  
pp. 119-123 ◽  
Author(s):  
D. M. Connolly ◽  
J. H. Williamson
Keyword(s):  
Measure Zero ◽  
General Context ◽  
Measure Algebra ◽  
Absolutely Continuous ◽  
Borel Measures ◽  
Perfect Set ◽  
Convolution Power ◽  
Recent Theorem ◽  
Vector Sum ◽  
Positive Results

The authors have recently treated (2) the problem of finding subsets E of the real line , of type Fσ, such that E–E contains an interval and the k-fold vector sum (k)E is of measure zero. Positive results can be obtained, for all k, on the basis of a recent theorem of J. A. Haight (3), following earlier partial results (1), (4) for k ≦ 7; and indeed in these cases the problem has a solution with E a perfect set. An analogous problem, apparently in most respects subtler than the first, is the following. Do there exist finite regular Borel measures μ on such that is absolutely continuous (where is the adjoint of μ) and the kth convolution power μk is singular? Both problems are of interest in the general context of elucidating the properties of the measure algebra or, more generally, M(G) for locally compact abelian G. The second problem may be regarded as an attempt to provide (at least one aspect of) a multiplicity theory for the first.

scholarly journals Fourier-Stieltjes Transforms which vanish at infinity off certain sets

1978 ◽  
Vol 19 (1) ◽  
pp. 49-56 ◽  
Author(s):  
Louis Pigno
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Measure Zero ◽  
Absolutely Continuous ◽  
Character Group ◽  
Convolution Algebra ◽  
Borel Measures ◽  
Discrete Measures ◽  
Stieltjes Transforms

In this paper G is a nondiscrete compact abelian group with character group Г and M(G) the usual convolution algebra of Borel measures on G. We designate the following subspaces of M(G) employing the customary notations: Ma(G) those measures which are absolutely continuous with respect to Haar measure; MS(G) the space of measures concentrated on sets of Haar measure zero and Md(G) the discrete measures.

scholarly journals A characterisation of a certain class of convolution algebras

1973 ◽  
Vol 18 (3) ◽  
pp. 198-198 ◽  
Author(s):  
J. W. Baker
Keyword(s):  
Compact Semigroup ◽  
Measure Algebra ◽  
Canonical Embedding ◽  
Absolutely Continuous ◽  
Borel Measures ◽  
Convolution Algebras ◽  
Image Position

In (6) Taylor has introduced the notion of a convolution measure algebra. In the same paper he constructed a canonical embedding of an arbitrary, semisimple commutative convolution measure algebra A into the algebra M(S) of all bounded, regular Borel measures on a compact semigroup S. This embedding has the properties that A is σ(M(S), C(S)-dense in M(S), that if μ is in A and ν is absolutely continuous with respect to μ, then ν is in A and that the set A^ of non-zero complex homomorphisms of A can be identified with the set S^ of continuous semicharacters of S where h ∈ A^ is identified with χ ∈ S^ by the equation

Embeddings of Müntz Spaces in

2019 ◽  
Vol 62 (1) ◽  
pp. 1-9
Author(s):  
Ihab Al Alam ◽  
Pascal Lefèvre
Keyword(s):  
Borel Measure ◽  
Essential Norm ◽  
Weak Compactness ◽  
Positive Borel Measure ◽  
Absolutely Continuous ◽  
Borel Measures ◽  
Embedding Operator ◽  
Müntz Spaces

AbstractIn this paper, we discuss the properties of the embedding operator $i_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6EC}}:M_{\unicode[STIX]{x1D6EC}}^{\infty }{\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707})$, where $\unicode[STIX]{x1D707}$ is a positive Borel measure on $[0,1]$ and $M_{\unicode[STIX]{x1D6EC}}^{\infty }$ is a Müntz space. In particular, we compute the essential norm of this embedding. As a consequence, we recover some results of the first author. We also study the compactness (resp. weak compactness) and compute the essential norm (resp. generalized essential norm) of the embedding $i_{\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}}:L^{\infty }(\unicode[STIX]{x1D707}_{1}){\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707}_{2})$, where $\unicode[STIX]{x1D707}_{1}$, $\unicode[STIX]{x1D707}_{2}$ are two positive Borel measures on [0, 1] with $\unicode[STIX]{x1D707}_{2}$ absolutely continuous with respect to $\unicode[STIX]{x1D707}_{1}$.

scholarly journals Small sets containing any pattern

2018 ◽  
Vol 168 (1) ◽  
pp. 57-73
Author(s):  
URSULA MOLTER ◽  
ALEXIA YAVICOLI
Keyword(s):  
Hausdorff Measure ◽  
Analogous Result ◽  
Measure Zero ◽  
General Construction ◽  
Dimension Function ◽  
Perfect Set ◽  
Finite Set ◽  
Isolated Points ◽  
Special Case

AbstractGiven any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set E in ${\mathbb{R}}^N$, of h-Hausdorff measure zero, such that for any finite set {f1,. . .,fn} ⊆ $\mathcal{F}$, E satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$.We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points.

Finite powers of strong measure zero sets

1999 ◽  
Vol 64 (3) ◽  
pp. 1295-1306 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Measure Zero ◽  
General Context ◽  
Partition Relation ◽  
Compact Metric Space ◽  
Totally Bounded ◽  
Zero Sets ◽  
Strong Measure ◽  
Strong Measure Zero

AbstractIn a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.

scholarly journals L-ideals of M(G) determined by continuity of translation

1973 ◽  
Vol 18 (4) ◽  
pp. 307-316 ◽  
Author(s):  
Gavin Brown ◽  
William Moran
Keyword(s):  
Abelian Group ◽  
Locally Compact Abelian Group ◽  
Norm Topology ◽  
Locally Compact ◽  
Total Variation Norm ◽  
Absolutely Continuous ◽  
Convolution Algebra ◽  
Borel Measures ◽  
Variation Norm ◽  
Stieltjes Transforms

G denotes a locally compact abelian group and M(G) the convolution algebra of regular bounded Borel measures on G. An ideal I of M(G) closed in the usual (total variation) norm topology is called an L-ideal if μ ∈ I, ν≪ μ (ν absolutely continuous with respect to μ) implies that ν ∈ I. Here we are concerned with the L-idealsL1(G), , and M0(G) where, as usual, L1(G) denotes the set of measures absolutely continuous with respect to Haar measure, denotes the radical of L1(G) in M(G) and M0(G) denotes the set of measures whose Fourier-Stieltjes transforms vanish at infinity.

Large families of singular measures having absolutely continuous convolution squares

1968 ◽  
Vol 64 (4) ◽  
pp. 1015-1022 ◽  
Author(s):  
Karl Stromberg
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Regular Measure ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Singular Measures ◽  
Large Families ◽  
Nikodym Derivative

In 1966, Hewitt and Zuckerman(3,4) proved that if G is a non-discrete locally compact Abelian group with Haar measure λ, then there exists a non-negative, continuous regular measure μon G that is singular to λ(μ ┴ λ) such that μ(G)= 1, μ * μ is absolutely continuous with respect to λ(μ * μ ≪ λ), and the Lebesgue-Radon-Nikodym derivative of μ * μ with respect to λ is in (G, λ) for all real p > 1. They showed also that such a μ can be chosen so that the support of μ * μ contains any preassigned σ-compact subset of G. It is the purpose of the present paper to extend this result to obtain large independent sets of such measures. Among other things the present results show that, for such groups, the radical of the measure algebra modulo the -algebra has large dimension. This answers a question (6.4) left open in (3).

Convolution in perfect Lie groups

2016 ◽  
Vol 161 (1) ◽  
pp. 31-45
Author(s):  
YVES BENOIST ◽  
NICOLAS DE SAXCÉ
Keyword(s):  
Hausdorff Dimension ◽  
Lie Groups ◽  
Haar Measure ◽  
Lie Group ◽  
Borel Measure ◽  
Absolutely Continuous ◽  
Borel Set ◽  
Open Set ◽  
Convolution Power ◽  
Smooth Density

AbstractLetGbe a connected perfect real Lie group. We show that there exists α < dimGandp∈$\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure onG, then thepth convolution power μ*pis absolutely continuous with respect to the Haar measure onG, with arbitrarily smooth density. As an application, we obtain that ifA⊂Gis a Borel set with Hausdorff dimension at least α, then thep-fold product setApcontains a non-empty open set.

On Intersections of Small Perfect Sets

2002 ◽  
Vol 9 (3) ◽  
pp. 495-505
Author(s):  
H. Fast
Keyword(s):  
Unit Circle ◽  
Measure Zero ◽  
Perfect Set ◽  
Perfect Subset

Abstract For a not empty perfect subset of the unit circle C there is a perfect subset of C measure zero which being rotated to every position intersects the first set on a nonempty perfect set. This result may be stated in terms of set of distances between pairs of points from these two sets. A generalization of this result to a product of tori is suggested.

Coin tossing and powers of singular measures

1975 ◽  
Vol 77 (2) ◽  
pp. 349-364 ◽  
Author(s):  
Gavin Brown ◽  
William Moran
Keyword(s):  
Lebesgue Measure ◽  
Probability Distributions ◽  
Main Question ◽  
Measure Algebra ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Singular Measures ◽  
Coin Tossing ◽  
Modern Apparatus ◽  
Interesting Account

We shall be concerned with the probability distributions which arise from naive coin tossing experiments and their repetitions. In each case our main question is whether the resulting measure is singular with respect to Lebesgue measure or absolutely continuous. It is a remarkable fact that althoughthese distributions were defined and studied some forty years ago (see (17)for an interesting account), they already provide examples necessary to recent studies of the convolution measure algebra of a locally compact abeliangroup. In order to verify this one is obliged to make substantial improvements on the classical results but, at least at the technical level, this requires no modern apparatus. Accordingly, we shall restrict attention to the circle. Before doing this, however, we summarize the motivational background from abstract harmonic analysis and indicate the applications of our results to that area.

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