Egoroff's Theorem Research Articles

Egoroff’s theorem and the distribution of standard points in a nonstandard model

We study the relationship between the Loeb measure °(*fi) of a set £ and the /?-measure of the set S(E) = [x\*x G E) of standard points in E. If E is in the a-algebra generated by the standard sets, then °(*f0(£) — piS(Ef). This is used to give a short nonstandard proof of Egoroffs Theorem. If £ is an internal, * measurable set, then in general there is no relationship between the measures of S(£) and E. However, if *X is an ultrapower constructed using a minimal ultrafilter on u, then *p(£) «i 0 implies that S(E) is a u-null set. If, in addition, y. is a Borel measure on a compact metric space and £ is a Loeb measurable set, then M(5(£)) R and /: X -» 7? is also a measurable function. Egoroffs Theorem [3] states 1.1 Egoroff's Theorem. 7/ S„ —* S pointwise almost everywhere then Sor every e > 0 there is a set A G R, and x G *X. Then x is said to be a point of intrinsic nonuniformity if there is an infinite integer v such that/,,(*) »*/(■*)• Let E denote the set of points of intrinsic nonuniformity. (Note: E is usually external.) 1.3 Definition. Suppose A is a (possibly external) subset of *X. A is said to have S-measure zero if for every standard e > 0 there is a standard set B G IS such that A G*B and y(B) ) [8, Theorem 4.6.1]. We need one more definition before proving Egoroffs Theorem. 1.5 Definition. Suppose A is a (possibly external) subset of *X. Let S(A) denote the set of all standard points in A. That is, S(A) = {x G X\*x G A). Note S(A) is just the standard part of A with respect to the discrete topology on X. 1.6 Proof of Egoroff's Theorem. Suppose/, —>/pointwise almost everywhere. Hence there is a set A G 6J such that y(A) = 0 and/„ -»/pointwise on X A. Let E denote the set of points of intrinsic nonuniformity. Then S(E) G A. Thus S(E) has measure zero and by II.3 E has S-measure zero, completing the proof by 1.4. II. The distribution of standard points in *X. The purpose of this section is to study the relationship between the measure (in a sense to be defined below) of a set E G*X and the standard measure of S(E). Intuitively, the standard points are evenly distributed in *X and one might, therefore, expect the measures of E and S(E) to be infinitely close for a reasonable class of sets E. II. 1 Definition. Let eE be the (external) algebra, & = {*A\A G k\Sr(x) *f(x) > I/«}Claim:

Egoroff's Theorem and the Distribution of Standard Points in a Nonstandard Model

Egoroff's Theorem and the Distribution of Standard Points in a Nonstandard Model

The range of a vector measure has the Banach-Saks property

The above result of Diestel and Seifert is proved using the Banach-Saks theorem for L p ( λ ) , 1 > p > ∞ {L^p}(\lambda ),1 > p > \infty .

Measure theoretic convergences of observables and operators

Definitions of different types of measure theoretic convergence for observables and operators are given. In particular we define convergence in measure, almost everywhere, everywhere, almost uniformly, and uniformly. These types of convergence are compared and characterized. Furthermore, our theory is compared to that of Segal-Stinespring. Convergence theorems such as a bounded convergence theorem, Fatou's lemma, and a special case of Egoroff's theorem are proved. We show that the general Egoroff's theorem does not hold.

A Remark on the Theorems of Lusin and Egoroff

In this note we do not intend to establish new results but only to suggest a very simple proof of Lusin's theorem, direct for σ-finite regular measures, a proof that bypasses the usual procedure of first establishing this theorem for sets of finite measure only. The proposed proof utilizes the notion of subuniform convergence, a method which seems not yet to have been used, despite its simplicity and adaptability. Simultaneously, a useful supplement to Egoroff's theorem will be obtained.

On Almost Uniform Convergence of Families of Functions

In [5] Tolstov showed by a counterexample that Egoroff' s theorem on almost uniform convergence cannot be extended to families of functions (ft(x)}, with t a continuous real parameter. However, Frumkin [2] proved that this is possible provided that some sets of measure zero (depending on t) are disregarded when each particular ft(x) is considered. This interesting result was obtained by using the rather involvec machinery of Kantorovitch' s semi-ordered spaces and Lp spaces. In the present note we intend to give a simpler and more general proof. Indeed, it will be seen that only a slight modification of the standard proof of Egoroff's theorem is necessary to obtain Frumkin' s theorem in a more general form. We shall establish the following result.

On Egoroff's theorem

On Egoroff's theorem

An alternative form of Egoroff's theorem

An alternative form of Egoroff's theorem

A Counter-Example Concerning Egoroff's Theorem

Journal of the London Mathematical SocietyVolume s1-34, Issue 2 p. 139-140 Notes and papers A Counter-Example Concerning Egoroff's Theorem Correction(s) for this article Addendum To a Note on Egoroffs Theorem J. D. Weston, Volume s1-35Issue 3Journal of the London Mathematical Society pages: 366-366 First Published online: December 23, 2016 J. D. Weston, J. D. Weston University of Durham, King's College Newcastle upon TyneSearch for more papers by this author J. D. Weston, J. D. Weston University of Durham, King's College Newcastle upon TyneSearch for more papers by this author First published: April 1959 https://doi.org/10.1112/jlms/s1-34.2.139Citations: 4AboutPDF 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 onFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volumes1-34, Issue2April 1959Pages 139-140 RelatedInformation