Conditional Measures and Applications.

Rao, M. M.

doi:10.2307/2533033

Cited by 41 publications

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“…Conditional measures and related problems from the theory of liftings are discussed in [26,51,52,53,81,82,194,195,202,203,219,248,297,314,316,322,324,325,326,327,334,360,367,376,384,411,413,417,420,441,460,482]. …”

Section: ~-~ (E)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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Section: ~-~ (E)mentioning

confidence: 99%

Measures on topological spaces

Богачев¹

1998

J Math Sci

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“…Þ is locally finite, we can make a disintegration by Rao, 1993). Since ðÄ; z; bÞ > 0 for all ðz; bÞ2NðÄÞ, it is possible to assume without loss of generality that, for ðz 1 ;…”

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confidence: 99%

On Second‐Order Characteristics of Germ‐Grain Models with Convex Grains

Ballani

2006

Mathematika

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“…By condition (E ′ ), µ n (|V |) = 0, and therefore the characteristic function of |V | equals 0 as an element of L 1 ([0, 1] n , µ n ). By the construction of the conditional measures (see, e.g., [12]), the set U ′ W of all u such that µ n u (|V | ∩ C u ) = 0 has ν measure 1. Since G 2 is a countable group and there exist countably many complexes W as above, the set of all u that satisfy both (i) and (ii) has ν measure 1.…”

Section: Proof Of Theorem 23mentioning

confidence: 99%

Invariant Measures in Free MV-Algebras

Panti

2008

Communications in Algebra

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Abstract. MV-algebras can be viewed either as the Lindenbaum algebras of Lukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MValgebra Freen is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0,1] n . The maximal spectrum of Freen is canonically homeomorphic to [0, 1] n , and the automorphisms of the algebra are in 1-1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this paper we prove that the only probability measure on [0, 1] n which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic logic, it means that the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure. PreliminariesAn MV-algebra is an algebra (A, ⊕, ¬, 0) such that (A, ⊕, 0) is a commutative monoid and the identities ¬¬f = f , f ⊕ ¬0 = ¬0, and ¬(¬f ⊕ g) ⊕ g = ¬(¬g ⊕ f ) ⊕ f are satisfied. MV-algebras can be viewed either as the Lindenbaum algebras of Lukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups (ℓ-groups) in which a strong order unit has been fixed. We recall that a strong unit in an ℓ-group G is a positive element u of G such that for every g ∈ G there exists a positive integer n for which g ≤ nu. The unit interval Γ(G, u) = {g ∈ G : 0 ≤ g ≤ u} is then an MV-algebra under the operations f ⊕ g = (f + g)∧u, ¬f = u − f , 0 = 0 G , and the functor Γ is an equivalence between the category of ℓ-groups with a distinguished strong unit and the category of MValgebras If A is viewed as the Lindenbaum algebra of some theory in Lukasiewicz logic, then m is a function assigning an "average truth-value" to the elements of A, i.e., to the propositional formulas modulo the theory.Key words and phrases. MV-algebras, state, automorphism-invariance, piecewise-linear homeomorphisms.2000 Math. Subj. Class.: 06D35; 37A05. The Γ functor induces a canonical bijection between the states of (G, u) and those of A = Γ(G, u) and a homeomorphism between the maximal spectrum of (G, u) and MaxSpec A. We recall that the maximal spectrum of (G, u) is the set of maximal ℓ-ideals of G, while MaxSpec A is the set of maximal ideals of A (i.e., kernels of homomorphisms from A to Γ(R, 1)); both sets are equipped with the Zariski topology [2, Chapitre 10]. We will formulate our results mainly in terms of MV-algebras, leaving to the reader their straightforward translation in the language of ℓ-groups with strong unit.The set of all states of A is a compact convex subset of [0,1] A , where the latter is given the product topology, and the subsp...

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Conditional Measures and Applications.

Cited by 41 publications

References 0 publications

Measures on topological spaces

Measures on topological spaces

On Second‐Order Characteristics of Germ‐Grain Models with Convex Grains

Invariant Measures in Free MV-Algebras

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