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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...
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...
Abstract.We introduce the notion of the weak density of a Boolean algebra and show that for homogeneous measure algebras it coincides with the density (=least size of a coinitial set). From this we obtain a partial liffîng of the measure algebra of [0, 1 ] of minimal size which does not extend to a lifting. It also follows that the ;r-character of each point and the ^-weight are the same for the Stone space of a homogeneous measure algebra 0. Notation For set-theoretic and topological notation not described below, see [3,8,9,10]. For measure theory see [7]. If B is a Boolean algebra and a e B , ac is the complement of a . Ba = B\a = {b € B: b < a} . B is homogeneous if Ba is isomorphic to B for all a e B\{0}. If / is a function which assigns an ordinal to each Boolean algebra, then B is homogeneous in f if f{Ba) = f(B) for all a 6 B\{0}. If (X,~L,n) is a probability space and A, B eZ then we write A ç* B, A =* B to mean ß(A\B) = 0, ß(AAB) -0, respectively. If / is any nonempty set, then (X ,//) is the product space with product measure denoted also by fi. For a e / let na: X1 -* XAM be defined by nJJ) = /|A{a} . Then for any set E c X , supp(£) == {a € /: E £ n~ n"aE}.For cardinals k > co, ß is the usual product measure on {0,1}*, ZK is the cr-algebra of measurable sets, and ¿VK = {E e ZK : ¡iE = 0} . srfK = E^/yT is the measure algebra of ({0,1}K ,2X ,/u). If E e ZK then [E] e sfK is the equivalence class of E. If B is a subalgebra of stfK, then a partial lifting p: B -> Z^ is a Boolean homomorphism satisfying />(a) e a for each a € B .
We investigate the cofinality of the partial order κ of functions from a regular cardinal κ into the ideal of Lebesgue measure zero subsets of R. We show that when add () = κ and the covering lemma holds with respect to an inner model of GCH, then cf (κ) = max{cf(κκ), cf([cf()]κ)}. We also give an example to show that the covering assumption cannot be removed.
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