A measure space without the strong lifting property

Losert, Viktor

doi:10.1007/bf01420369

Cited by 22 publications

(9 citation statements)

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“…In [314], V. Losert constructed his famous counterexample. It was open for a long time whether the metrizability assumption could be dropped.…”

Section: B (635)mentioning

confidence: 99%

Measures on topological spaces

Богачев¹

1998

J Math Sci

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IntroductionIntegration on topological spaces is a field of mathematics which could be defined as the intersection of functional analysis, general topology, and probability theory. However, at different epochs the roles of these three ingredients were different, and, moreover, very often none of the three exerted a dominating'influence. For example, the theory of topological groups and analysis on manifolds gave rise to questions concerning Haar measures, Riemannian volumes, and other measures on locally compact spaces, and their influence was so strong that until recently many fundamental books on integration dealt exclusively with locally compact spaces. On the other hand, quantum fields and statistical physics provide problems of a totally different type, and this circumstance results in another trend in the theory of integration. At present, measure theory is especially strongly influenced by the intensive developmdnt of infinite-dimensional analysis in a broad sense, including stochastic analysis, dynamic systems, and the theory of representations of groups. This development involves measures on complicated infinite-dimensional manifolds and functional spaces. Recent investigations in population genetics have given rise to measure-valued diffusions, which, in turn, lead to such objects as measures on spaces of measures.The main aim of this survey is to present a systematic exposition of the integration theory on topological spaces, having in mind the indicated tendencies. Therefore, the target readership includes topologists, functional analysts, probabilists, and mathematical physicists. However, the main accent is put on analytical and probabilistic aspects rather than on purely topological or set-theoretic concepts. For this reason, no special topological knowledge is assumed.There are several books and recent surveys presenting modern measure theory on topological spaces. Schwartz' book [457] remains a standard reference book for basic questions. The results obtained after the publication of this book and many more special issues with a strong emphasis on general topology are discussed in excellent survey papers [183, 185] and [540]. A good introduction to the whole direction is given in Chapter 1 of the very informative monograph [527]. In addition, there are a number of books on gene:al measure theory, functional analysis, and probability theory, which include material on integration on topological spaces (the corresponding references are given in the text). However, there is no systematic exposition of modern theory oriented foward nontopologists. In addition, not all aspects of tile theory which are important for applications (in particular, those mentioned above) have been discussed in the literature. The central topics of this survey are:(i) regularity properties of measures on general topological spaces and specific spaces that arise in probability theory and functional analysis, including extension theorems for measures, (ii) transformations of measures and related problems such as conditional me...

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“…In [314], V. Losert constructed his famous counterexample. It was open for a long time whether the metrizability assumption could be dropped.…”

Section: B (635)mentioning

confidence: 99%

Measures on topological spaces

Богачев¹

1998

J Math Sci

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“…Fremlin's simplification of Losert's [18] celebrated counter-example to the strong lifting conjecture gives such a system. Indeed, let µ be Fremlin's Radon probability measure on X := {0, 1} ℵ 2 which has no strong lifting and is supported by X (cf.…”

Section: R E M a R K S (I)mentioning

confidence: 99%

On strong liftings for projective limits

Macheras

Strauss²

1994

Fund. Math.

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Abstract. We discuss the permanence of strong liftings under the formation of projective limits. The results are based on an appropriate consistency condition of the liftings with the projective system called "self-consistency", which is fulfilled in many situations. In addition, we study the relationship of self-consistency and completion regularity as well as projective limits of lifting topologies.Introduction. Only recently in [19] the general theory of inductive limits of (topological) measure spaces was developed by N. D. Macheras and at the same time the permanence of strong lifting was established for inductive limits. For much longer time the projective limit of measure spaces is in common use (see e.g. Bochner [4], Choksi [5], Musia l [25], Rao [29]) but there seems to be no discussion of the permanence of strong liftings for general projective limits. Only for finite or countable products there exist permanence results in the forthcoming paper [22].As in [22] our main concern in this paper is with conditions of compatibility of the liftings in the factors with the lifting on the limit or product. The notion of the "consistent lifting" of M. Talagrand [30] seems to be the first example on this line. Talagrand's paper is only for finite products in which all factors must be equal. [22] gives consistency conditions for finite and countable products with different factors, and our basic sufficient condition for the existence of strong liftings on projective limits in this paper, the so-called "self-consistency" (see Section 1 for definitions), may be read as a strengthening of Talagrand's consistent lifting, i.e. to be precise in his special instance it is a condition in terms of the generators of the product σ-algebra while ours is a condition on the whole σ-algebra. Remark 2.2(iii) gives a list of projective systems which allow self-consistent liftings. Among them are always countable systems provided all factors have the universally strong lifting property (USLP for short). The basic existence result for For products there is a well-known but somewhat elusive relationship between completion regularity and the existence of strong liftings (see e.g.[6]). By Theorem 3.1, self-consistency is sufficient (but again not necessary by Remark 3.3) for the permanence of completion regularity under projective limits of compact spaces and more generally for Hausdorff completely regular spaces in the presence of sequential maximality. Corollary 3.2 gives a permanence result for strong Baire liftings in projective limits.In Section 4 we study the projective limit for lifting topologies. In terms of lifting topologies equivalent conditions can be given for the existence of a strong lifting on the projective limit (see Theorem 4.1 and its corollaries). Section 5 contains the specialization to products, thus extending the results in [22] from countable to uncountable products (see Theorem 5.3).One consequence of the basic result (Theorem 5.3) is the existence of a strong lifting if all factors have the USLP (see Theorem...

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“…Clearly this is not true in general, since measure compactness does not even imply lifting compactness, a condition which lies strictly between strong measure compactness and measure compactness (see A. Bellow [2]). V. Losert's counterexample in [11] together with 2.3 shows that neither lifting compactness nor strong measure compactness imply strong lifting compactness. Nor does strong lifting compactness imply strong measure compactness, as witnessed by the standard Lebesgue non-measurable subset of [0,1].…”

Section: Strongly Lifting Compact Spacesmentioning

confidence: 99%

On strong lifting compactness, with applications to topological vector spaces

1986

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The notion of strong lifting compactness is introduced for completely regular Hausdorff spaces, and its structural properties, as well as its relationship to the strong lifting, to measure compactness, and to lifting compactness, are discussed. For metrizable locally convex spaces under their weak topology, strong lifting compactness is characterized by a list of conditions which are either measure theoretical or topological in their nature, and from which it can be seen that strong lifting compactness is the strong counterpart of measure compactness in that case.

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A measure space without the strong lifting property

Cited by 22 publications

References 6 publications

Measures on topological spaces

Measures on topological spaces

On strong liftings for projective limits

On strong lifting compactness, with applications to topological vector spaces

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