Type: Article
Publication Date: 1952-12-01
Citations: 9
DOI: https://doi.org/10.2140/pjm.1952.2.531
Introduction.We consider the problem of defining a nontrivial, translation-invariant Borel measure over real separable Hubert space.As noted by Loewner [4l, this is not possible; but instead of relinquishing as he does the real number system for a non-Archimedean ordered field for the values of a "measure,*' we shall consider several topological subspaces of Hubert space arising frequently in analysis.These are locally compact; and using either the Kolmogoroff stochastic processes construction [2], or else following the Haar measure construction [ l] or L 5], we can get a nontrivial, essentially translationinvariant Borel measure.However, since the special subspaces considered are not groups under translation, and do not even contain a group germ, the usual Haar measure construction must be modified in a special fashion, and the precise translation invariance obtained is somewhat restrictive.Actually we carry through this modified Haar measure construction for the more general situation of a locally compact translation space, which is defined as an appropriate subspace of an Abelian topological group.The results are collected in a summary at the end. Formulation of the problem. Letthe square summable real sequences and thus the real separable Hubert space prototype.Since "ί 2 i s a subset of Roc, the countably infinite Cartesian product of the real line ( -oo, oo), we have available on Λs 2 a s we ll as the ΛJ 2 norm metric topology also the product topology defined relatively from Roc Under these two topologies we shall consider the Έ 2 -subsets X = 1 x C ^2 I I x n I <.h(n) for all n },