Extensions of regular lattice measures with topological applications

Bachman, George; Sultan, Alan

doi:10.1016/0022-247x(77)90245-1

Journal of Mathematical Analysis and Applications

1977

DOI: 10.1016/0022-247x(77)90245-1

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Extensions of regular lattice measures with topological applications

George Bachman

Alan Sultan

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Cited by 23 publications

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“…Introduction* In earlier papers [6], [7], [49], we have developed for an abstract set X and a given lattice £f of subsets, the concept of i^-repleteness. This concept, as well as others such as incompact, iίP-countably compact, etc., considered by Alexandroff [1][2][3], Meyer [41], Marczewski [39], Tops0e [50], Frolik [19], and others, we have expressed measure theoretically in terms of two valued .Sf-regular measures.…”

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confidence: 99%

“…The notion of ^-replete includes as special cases: real compact, Borel complete [24], α-complete [15], etc., and in [6], we developed measure-theoretic results to show systematically how to obtain repleteness interrelations. Here, we are concerned with mapping and subspace problems.…”

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“…( [51], p. 170)). General properties and representation theorems for countably compact lattices can be found in [6], (14) £? is called regular if for A e £f, and xίA, there exist B,Ce^f such that xeB', AczC, and B' Π C = 0.…”

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“…We will This theorem generalizes many specific topological type regularity theorems in the literature. For the details of the proof and applications see [6]. For the specific case of zero-one measures see [7].…”

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Regular lattice measures: mappings and spaces

Bachman¹,

Sultan²

1976

Pacific J. Math.

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We prove in this paper a very general measure extension theorem which has as corollaries many recent, significant extension theorems in the literature. We apply these results to the question of when there is a well behaved map from the σ-smooth lattice regular measures on one set to the σ-smooth lattice regular measures on a second set. After developing these general theorems we specialize consideration to two valued latitce regular measures and obtain in a new and consistent manner many important mapping and subspace theorems on the preservation of different types of repleteness including results of Dykes, Hager, Isiwata, Moran, Varadarajan, Gillman, Jerrison and others. Frolik [19], and others, we have expressed measure theoretically in terms of two valued .Sf-regular measures. This is advantageous; for besides being analytically simpler to work with than with filters, many theorems in this form can be generalized naturally to arbitrary irregular measures, and the topological settings extended from Wallman topologies to vague topologies.The notion of ^-replete includes as special cases: real compact, Borel complete [24], α-complete [15], etc., and in [6], we developed measure-theoretic results to show systematically how to obtain repleteness interrelations. Here, we are concerned with mapping and subspace problems. The mapping questions are of the type: Given T: X-• Y which is well-behaved (see §5 for details) with respect to two lattices £έ\ y J^ of subsets of X and Y respectively, when does T induce a well-behaved mapping T**: MR{σ, £? ^-*MR{σ, & §> where, in general, MR(σ, £f) designates the σ-smooth i5f-regular measures on a set X with respect to a lattice, ^ of subsets? We develop the major results of this type in §5, and when applied to the special case of two-valued measures, we get as corollaries important subspace and mapping results of Frolik [21]

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Regular lattice measures: mappings and spaces

Bachman¹,

Sultan²

1976

Pacific J. Math.

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“…We do this first in some special cases. 4.1 Theorem. // X is a locally compact T2 space and if Bx = B2 = the collection of continuous functions of compact support, <f> = <£>*, £, = SCg, and £2 = ÍF, then v extends p. Thus in a locally compact T2 space every p E MR (iKn) extends to a v E MR (a, 5F).…”

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A General Measure Extension Procedure

Sultan¹

1978

Proceedings of the American Mathematical Society

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Abstract. We give a very general and flexible way of producing measure extensions. We obtain as corollaries many well-known and important measure extension and integral representation theorems as well as the main theorems of several recent papers.Introduction. The question of when one can extend a measure (finitely, countably, or arbitrarily additive) is a very old one. Many different procedures over the years have been applied to obtain different types of significant extension theorems, mostly in topological spaces. Many of these have found numerous important applications in diverse branches of mathematics. Some of the more recent extension theorems have found applications in solving topological questions and are giving a great deal of insight into the interplay between measure and topology (see, in particular, [4], [5], [6]). In this paper we give a single, unifying procedure, which allows one to get simultaneously many of the well-known measure extension theorems. The method presented here is particularly useful for a variety of reasons: (1) It lends itself immediately to applications in the finite or totally finite cases. (2) It gives a great deal of flexibility in the actual construction of the measure extension. (3) In most of the important cases where the extension of the measure is not unique, this procedure gives all possible ways of extending the given measure. (4) The method ties in very naturally with many well-known representation theorems, and one obtains many of these as corollaries. (5) The actual construction of the measure extension is simple.In the very simplest cases we obtain many nontrivial extension theorems for locally compact T2 spaces. When our results are applied appropriately in many of the other cases, we obtain the main theorems of Definitions. By a paving (X, E) we mean a set X together with a lattice £ of subsets of X. We will suppress X and just say that £ is a paving. If 0 E £

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Extensions of group‐valued regular Borel measures

Khurana

1980

Mathematische Nachrichten

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In ([ill) a result is proved about the extension of regular BOREL measures. The main result isTheorem ([Ill, Theorem 10). Let X and Y be compact HAUSDORFF spaces, 9 : X -. Y a continuous onto-mapping and po a non-negative regular BOREL measure on Y. Then there exists a non-negative regular BOREL memure p on X such that p ( A ) = p 0~-1 ( A ) ) ,Theorem 9, easily reduces to this theorem.) This theorem is a simple consequence of the HAHN-BANACH theorem ([4], [lo], [9]). First we make some remarks about notations. N will always be the set of natural numbers. For a compact HAUSDORFF space X , C ( X ) will denote the set of all real-valued continuous functions on X , a ( X ) the set of all BOREL suhsets of X, B o ( X ) all bounded, real-valued BOREL measurable functions on X , B ( X ) all bounded realvalued functions on X, and llhIl=sup I h (~) j , for any hE R o ( X ) . I n a vector lattice E order-convergence, order-continuity, order o-continuity, etc., are taken in the sense of ([?I, p. 184; see also [13], [IS]). B a ( X )will denote the set of all bounded, real-valued, RAIRE measurable functions on X. A11 locally convex spaces are taken on the field of real numbers. &,(X) denotes all BAIRE subsets of X.In thjs paper we shall prove some extensions of this result. With the above notations we shall prove the following result.

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Extensions of regular lattice measures with topological applications

Cited by 23 publications

References 29 publications

Regular lattice measures: mappings and spaces

Regular lattice measures: mappings and spaces

A General Measure Extension Procedure

Extensions of group‐valued regular Borel measures

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