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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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2017/9853672 |
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| _version_ | 1850176325608800256 |
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| author | Jean-Pierre Magnot |
| author_facet | Jean-Pierre Magnot |
| author_sort | Jean-Pierre Magnot |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b0bf486cdb6a4254aa61e0dd382accf8 |
| institution | OA Journals |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-b0bf486cdb6a4254aa61e0dd382accf82025-08-20T02:19:16ZengWileyJournal of Mathematics2314-46292314-47852017-01-01201710.1155/2017/98536729853672The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue MeasuresJean-Pierre Magnot0Lycée Jeanne d’Arc, Avenue de Grande Bretagne, 63000 Clermont-Ferrand, FranceOne 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.http://dx.doi.org/10.1155/2017/9853672 |
| spellingShingle | Jean-Pierre Magnot The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures Journal of Mathematics |
| title | The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures |
| title_full | The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures |
| title_fullStr | The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures |
| title_full_unstemmed | The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures |
| title_short | The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures |
| title_sort | mean value for infinite volume measures infinite products and heuristic infinite dimensional lebesgue measures |
| url | http://dx.doi.org/10.1155/2017/9853672 |
| work_keys_str_mv | AT jeanpierremagnot themeanvalueforinfinitevolumemeasuresinfiniteproductsandheuristicinfinitedimensionallebesguemeasures AT jeanpierremagnot meanvalueforinfinitevolumemeasuresinfiniteproductsandheuristicinfinitedimensionallebesguemeasures |