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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| Vydáno v: | Journal of Mathematics Ročník 2017; číslo 2017; s. 1 - 14 |
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| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cairo, Egypt
Hindawi Publishing Corporation
01.01.2017
Hindawi John Wiley & Sons, Inc Wiley |
| Témata: | |
| ISSN: | 2314-4629, 2314-4785 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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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| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 |
| ISSN: | 2314-4629 2314-4785 |
| DOI: | 10.1155/2017/9853672 |