Projective limits of probability spaces

Rao, M. M.

doi:10.1016/0047-259x(71)90028-5

Cited by 30 publications

(18 citation statements)

References 10 publications

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“…The general case in which the P n are probability distributions is a difficult mathematical problem (Rao [19]), much studied in the Bayesian literature, for example in connection with 'hierarchies of beliefs' (see Brandenburger and Dekel [4] for an interesting example). In order for the limit lim n→∞ P n to exist for all events, a complicated coherence condition must be satisfied.…”

Section: Probability and Uncertaintymentioning

confidence: 99%

The logical response to a noisy world

Stenning¹,

Lambalgen²

2010

Cognition and ConditionalsProbability and Logic in Human Thinking

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Section: Probability and Uncertaintymentioning

confidence: 99%

The logical response to a noisy world

Stenning¹,

Lambalgen²

2010

Cognition and ConditionalsProbability and Logic in Human Thinking

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“…-(a-) compact. Note that many authors use the term 'compact' as a synonym for K0-compact (e.g., [9]). _ Given {X ,3 § ,P), denote by f% the completion of 3 § with respect to P ;…”

Section: Statement Of Resultsmentioning

confidence: 99%

Compact Measures Have Loeb Preimages

Ross¹

1992

Proceedings of the American Mathematical Society

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Abstract.A compact measure is a (possibly nontopological) measure that is inner-regular with respect to a compact family of measurable sets. The main result of this paper is that every compact probability measure is the image, under a measure-preserving transformation, of a Loeb probability space. This generalizes a well-known result about Radon topological probability measures. It is also proved that a compact probability space can be topologized in such a way that the measure is essentially Radon. IntroductionWhich probability measures can be represented by a Loeb space? In [2] Robert Anderson proved that every Radon probability measure on a Hausdorff space X is the image, under the standard part map, of a Loeb measure on *X. Depending on the topology on X, a converse can be proved; for example, D. Landers and L. Rogge [4] prove: if X is regular, then every probability measure, which is the image under the standard part map of a Loeb measure on *X, is Radon.The situation when X is not a Hausdorff topological space is more problematic, as the standard part map is not defined. An early approach to representing nontopological probability spaces by the Loeb measure [5] dispenses with measurable transformations altogether, but does not appear to have much application. The case when X is topological but not Hausdorff would seem to be easier than the general case, but still suffers from the lack of a standard part map (although T. Norberg [7] has recently constructed a natural analogue of the standard part map for so-called 'sober' spaces). Such spaces arise naturally in extremal theory and the theory of random closed sets (see e.g., [8] or [11]).In this paper I consider a nontopological analogue of a Radon space, called a compact probability space. The main theorem is that every compact probability space is the image, under a measure-preserving transformation cp , of a Loeb space, cp is very easy to construct, and acts very much like the standard part map; in particular, it is used to define a topology on X with respect to which cp is the standard part map, provided this topology is Hausdorff.

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“…Proof -Let w be an element of Π. Let us look at the effect of replacing for exampleg 1 bỹ g 1 w in (14). Set F 1 = L(e 1 ) and F 2 = L(e −1 1 ).…”

Section: Most Of the Following Proposition Consists In Definitionsmentioning

confidence: 99%

“…According to Proposition 3.1 and general results on projective limits of measure spaces (see [14]), it is possible to take the projective limit of the compact Borel probability spaces (G E + , P G T,z ) G∈G with respect to the mappings (f GG ′ ) G≤G ′ .…”

Section: Random Holonomy Along Piecewise Geodesic Pathsmentioning

confidence: 99%

Discrete and continuous Yang-Mills measure for non-trivial bundles over compact surfaces

Lévy

2006

Probab. Theory Relat. Fields

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In this paper, we construct one Yang-Mills measure on a compact surface for each isomorphism class of principal bundles over this surface. For this, we refine the discretization procedure used in a previous construction [8] and define a new discrete theory which is essentially a covering of the usual one. We prove that the measures corresponding to different isomorphism classes of bundles or to different total areas of the base space are mutually singular. We give also a combinatorial computation of the partition functions which relies on the formalism of fat graphs.

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Projective limits of probability spaces

Cited by 30 publications

References 10 publications

The logical response to a noisy world

The logical response to a noisy world

Compact Measures Have Loeb Preimages

Discrete and continuous Yang-Mills measure for non-trivial bundles over compact surfaces

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