Limits of inverse systems of measures

Mallory, J. D. ; Sion, Maurice

Annales de l'Institut Fourier, Tome 21 (1971), p. 25-57 / Harvested from Numdam

Résumé

Étant donné un système projectif d’espaces mesurés $(X_i,\mu _i)_{i\in I}$, on étudie le problème d’existence d’une limite projective en considérant d’abord une mesure $\tilde{\mu }$ définie sur le produit $\prod _{i\in I}X_i$. Sous de simples conditions de régularité des $\mu _i$, on montre que $\tilde{\mu }$ a presque toutes les propriétés d’une limite. En outre, la limite projective $\mu ^{\prime }$ peut exister seulement si $\tilde{\mu }$ est elle-même une “limite” dans un sens plus général et $\mu ^{\prime }$ est alors la restriction de $\tilde{\mu }$ à l’ensemble limite des $X_i$. On obtient des résultats plus forts que ceux connus jusqu’à présent en examinant cette restriction.

Abstract

In this paper the problem of the existence of an inverse (or projective) limit measure $\mu ^{\prime }$ of an inverse system of measure spaces $(X_i,\mu _i)$ is approached by obtaining first a measure $\tilde{\mu }$ on the whole product space $\prod _{i\in I}X_i$.

The measure $\tilde{\mu }$ will have many of the properties of a limit measure provided only that the measures $\mu _i$ possess mild regularity properties.

It is shown that $\mu ^{\prime }$ can only exist when $\tilde{\mu }$ is itself a “limit” measure in a more general sense, and that $\mu ^{\prime }$ must then be the restriction of $\tilde{\mu }$ to the projective limit set $L$.

Results stronger than those previously known are obtained by examining $\tilde{\mu }$ restricted to $L$.

Détail

Publié le : 1971-01-01

MR 44 #1782 | Zbl 0205.07101

DOI : https://doi.org/10.5802/aif.361

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