The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures

One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of mea...

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Published in:Journal of Mathematics Vol. 2017; no. 2017; pp. 1 - 14
Main Author: Magnot, Jean-Pierre
Format: Journal Article
Language:English
Published: Cairo, Egypt Hindawi Publishing Corporation 01.01.2017
Hindawi
John Wiley & Sons, Inc
Wiley
Subjects:
Dimensional measurements
Functions (mathematics)
Heuristic
Hilbert space
Infinite
Infinity
Integrals
Invariants
Mathematical analysis
Mathematical research
Mathematics
Mean value theorems (Calculus)
Measure theory
Translations
ISSN:2314-4629, 2314-4785
Online Access:Get full text
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Abstract One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of means. In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (e.g., on a Hilbert space) is defined. This last object, which is not the classical infinite dimensional Lebesgue measure but its “normalized” version, is shown to be invariant under translation, scaling, and restriction.
AbstractList One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of means. In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (e.g., on a Hilbert space) is defined. This last object, which is not the classical infinite dimensional Lebesgue measure but its "normalized" version, is shown to be invariant under translation, scaling, and restriction.
Audience Academic
Author Magnot, Jean-Pierre
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CitedBy_id crossref_primary_10_3390_math11051161
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Cites_doi 10.4064/sm-7-1-1-15
10.1007/bf00994909
10.2307/2374298
10.1088/1742-6596/410/1/012003
10.1002/cpa.3160270205
10.1016/j.bulsci.2004.05.005
10.1016/j.crma.2003.11.022
10.1016/j.jfa.2004.07.014
10.1090/s0002-9939-04-07372-1
ContentType Journal Article
Copyright Copyright © 2017 Jean-Pierre Magnot.
COPYRIGHT 2017 John Wiley & Sons, Inc.
Copyright © 2017 Jean-Pierre Magnot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright_xml – notice: Copyright © 2017 Jean-Pierre Magnot.
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SubjectTerms Dimensional measurements
Functions (mathematics)
Heuristic
Hilbert space
Infinite
Infinity
Integrals
Invariants
Mathematical analysis
Mathematical research
Mathematics
Mean value theorems (Calculus)
Measure theory
Translations
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Title The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures
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