On an Invariant Borel Measure in Hilbert Space
Volume 52 / 2004
Bulletin Polish Acad. Sci. Math. 52 (2004), 47-51
MSC: 28A35, 28C20, 60G15.
DOI: 10.4064/ba52-1-5
Abstract
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].
