On an Invariant Borel Measure in Hilbert Space

G. Pantsulaia¹

¹ Department of Mathematics Georgian Technical University 77 Kostava St. 0175 Tbilisi 75, Georgia

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) no. 1, pp. 47-51.

Résumé

An example of a nonzero $\sigma $-finite Borel measure $\mu $ with everywhere dense linear manifold ${{\Bbb I}}_{\mu }$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space $\ell _2$ such that $\mu $ and any shift $\mu ^{(a)}$ of $\mu $ by a vector $a \in \ell _2 \setminus {{\Bbb I}}_{\mu }$ are neither equivalent nor orthogonal. This extends a result established in [7].

Détail

DOI : 10.4064/ba52-1-5

Keywords: example nonzero sigma finite borel measure everywhere dense linear manifold bbb admissible sense invariance translation vectors constructed hilbert space ell shift vector ell setminus bbb neither equivalent nor orthogonal extends result established