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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Bibliographic Details
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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Summary: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.
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ISSN:2314-4629
2314-4785
DOI:10.1155/2017/9853672