Sets Of Infinite Measure Research Articles

Discrete uniqueness sets for functions with spectral gaps

It is well known that entire functions whose spectrum belongs to a fixed bounded set admit real uniformly discrete uniqueness sets. We show that the same is true for a much wider range of spaces of continuous functions. In particular, Sobolev spaces have this property whenever is a set of infinite measure having `periodic gaps'. The periodicity condition is crucial. For sets with randomly distributed gaps, we show that uniformly discrete sets satisfy a strong non-uniqueness property: every discrete function can be interpolated by an analytic -function with spectrum in . Bibliography: 9 titles.

On grand Lebesgue spaces on sets of infinite measure

On grand Lebesgue spaces on sets of infinite measure

On the convergence of series in spaces of integrable functions

A sufficient condition for the convergence of series in the spaces Lp on a set of infinite measure is obtained.

A Utility Representation for a Preference Relation on a @s-Algebra

MODELS OF ECONOMIES with land used by urban economists often employ assumptions that are inappropriate for the commodity known as land (see Berliant [3]). It is therefore natural to attempt to formulate a model that does not suffer from this drawback. In particular, a framework with a finite number of consumers trading in measurable subsets of land (rather than densities or points) is constructed below. The purpose of the present study is to examine the preference relations and utilities used in this type of framework. Problems arise in generating utility representations of preference relations because the separability property of a oX-algebra (on a set of possibly infinite measure) endowed with the L' topology on indicator functions of elements of the a-algebra is not obvious. Let (L, A, m) be a nonatomic measure space with L a separable metric space, A its Borel v-algebra, and m regular and positive. Generally, script letters will represent subsets of $ while capital letters will represent elements of $. If A Be , define A/B={xc L xc A, xiB}. For example, L could be a measurable subset of R2, $ the c-algebra of all measurable subsets of L, and m Lebesgue measure restricted to L. The set L is then a representation of land in that anything immobile can be embedded in L, and consumers have complete preorderings over these immobile objects along with land through preference relations over subsets of L. There is no linear structure imposed on A. There are two reasons to consider all sets in A rather than only those with finite measure. First, it is natural to include L the totality of land, in the choice set of each agent of the example. Second, it is convenient to embed indicator functions of elements of $ in L' (see Bewley [6]), and completeness of the space requires that sets of infinite measure be included. Berliant [4] uses utility functions on L' to prove existence of an equilibrium in an economy with land. Next, a topology is imposed on A. Let the basis for the topology be given by T={$2cAi j$9={Bc$X 3jAcBccC m(B\A)>O wm(C\B)>O} for A, Cc X, Ac C, m(C\A)>O, m(A) O} for Cc$A open}u{$c9X |2 ={Bc$91| A'B. m(B\A)> O} for A c $ compact, m(A) <oo}. Thus, basis elements consist of subsets of $ that can be trapped (in a set containment sense) between various

Strongly equivalent invariant measures

AbstractTwo measures are strongly equivalent if they have the same sets of zero measure and the same sets of infinite measure. Given a group G of strongly non-singular measurable transformations of a non-atomic positive measure space (X, β, p), if G is amenable, then a necessary and sufficient condition for there to be a G-invariant positive measure on (X, β) which is strongly equivalent to p is that p(E) &gt; 0 implies inf p(gE) &gt; 0 and also p(E) &lt; ∞ implies

A set of infinite measure whose ratio set does not contain a given sequence

MathematikaVolume 22, Issue 2 p. 195-201 Research Article A set of infinite measure whose ratio set does not contain a given sequence J. A. Haight, J. A. Haight Department of Mathematics, University College, LondonSearch for more papers by this author J. A. Haight, J. A. Haight Department of Mathematics, University College, LondonSearch for more papers by this author First published: 01 December 1975 https://doi.org/10.1112/S0025579300006070Citations: 8AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume22, Issue2December 1975Pages 195-201 RelatedInformation

Analysis of tangential properties of curves of infinite length

In a recent paper (1) I have studied tangential properties of sets of infinite measure. This note represents a complementary study of sets and, in particular of arcs, of σ-finite linear measure. Suppose we have a linearly measurable set E of infinite measure that can be represented as the sum of sets of finite measure. Write En = En, 1 + En, 2, where En, 1 is the set of regular points of En and En, 2 that of irregular ones. The set F = ΣEn, 1 is a regular component of E and the set G = ΣEn, 2 an irregular one. A tangent toEn exists at almost all points of En, 1 and at almost no points of En, 2. Denote by Tn the set of all tangents to En and by T the sum-set ΣTn. T will be called a tangent-set to E. Lines of T are defined corresponding to almost all points of F and to the points of a subset of G of measure zero.

Non σ-finite closed subsets of analytic sets

A set is said to be σ-finite (with respect to Λs-measure) if it can be expressed as a countable sum of sets of finite Λs-measure. I have proved(1) that every non σ-finite analytic set in a Euclidean space contains a closed set of infinite measure, and Prof. Besicovitch asked me whether the closed subset could itself be chosen to be non σ-finite. In this paper an affirmative answer is given.